Content-Type: multipart/mixed; boundary="-------------9910290314978" This is a multi-part message in MIME format. ---------------9910290314978 Content-Type: text/plain; name="99-410.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-410.keywords" correlation, semi-classical. ---------------9910290314978 Content-Type: application/x-tex; name="Highordcorr.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Highordcorr.tex" \magnification=1200 \overfullrule =0pt \def\l{\ell} \def\parno{\par \noindent} \def\ref#1{\lbrack {#1}\rbrack} \def\leq#1#2{$${#2}\leqno(#1)$$} \def\vvekv#1#2#3{$$\leqalignno{&{#2}&({#1})\cr &{#3}\cr}$$} \def\vvvekv#1#2#3#4{$$\leqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr}$$} \def\vvvvekv#1#2#3#4#5{$$\leqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr &{#5}\cr}$$} \def\ekv#1#2{$${#2}\eqno(#1)$$} \def\eekv#1#2#3{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr}$$} \def\eeekv#1#2#3#4{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr}$$} \def\eeeekv#1#2#3#4#5{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr &{#5}\cr}$$} \def\eeeeekv#1#2#3#4#5#6{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr &{#5}\cr &{#6}\cr}$$} \def\eeeeeekv#1#2#3#4#5#6#7{$$\eqalignno{&{#2}&({#1})\cr &{#3}\cr &{#4}\cr &{#5}\cr&{#6}\cr&{#7}\cr}$$} \def\cint{{1\over 2\pi i}\int} \def\iint{\int\hskip -2mm\int} \def\iiint{\int\hskip -2mm\int\hskip -2mm\int} \def\buildover#1#2{\buildrel#1\over#2} \font \mittel=cmbx10 scaled \magstep1 \font \gross=cmbx10 scaled \magstep2 \font \klo=cmsl8 \font\liten=cmr10 at 8pt \font\stor=cmr10 at 12pt \font\Stor=cmbx10 at 14pt \centerline{\Stor Complete asymptotics for correlations of } \centerline{\Stor Laplace integrals in the semi-classical limit.} \medskip \centerline{{\stor Johannes Sj\"ostrand}\footnote{*}{\noindent Centre de Math\'ematiques, Ecole Polytechnique, F--91128 Palaiseau cedex, and UMR 7640 of CNRS\smallskip Key words: correlation, semi-classical. 1991 subject classification: Primary 82B20, secondary: 81Q20}} \medskip \par\noindent \it Abstract. \rm\liten In this paper we study the exponential asymptotics of correlations at large distance associated to a measure of Laplace type. As in [S1], [BJS], we look at a semi-classical limit. While in those papers we got the exponential decay rates and the prefactor only up to some factor $(1+{\cal O}(h^{1/2}))$, where $h$ denotes the small semi-classical parameter, we now get full asymptotic expansions. The main strategy is the same as in the quoted papers, namely to use an identity ([HS]) involving the Witten Laplacian of degree 1, and a Grushin (Feshbach) reduction for the bottom of the spectrum of this operator. The essential difference is however that we now have to use higher order Grushin problems (amounting to the study of a larger part of the bottom of th spectrum). In a perturbative case, the strategy of higher order Grushin problems was recently implemented by W.M.Wang [W] to get a few terms in the perturbative expansion of the decay rate.\rm \smallskip \par\noindent \it Resum\'e. \rm\liten Dans cet article nous \'etudions l'asymptotique exponentielle des corr\'elations \`a grande distance associ\'ees \`a une mesure de type de Laplace. Comme dans [S1], [BJS], on se place dans la limite semi-classique. Dans ces travaux nous avons obtenu le taux de d\'ecroissance exponentielle et le pr\'efacteur \`a un facteur $(1+{\cal O}(h^{1/2}))$ pr\`es, o\`u $h$ d\'esigne le petit param\`etre semi-classique. Nous obtenons maintenant des d\'eveloppements asymptotiques compl\`ets. La strat\'egie est la m\^eme que dans [S1,BJS]: Utiliser une identit\'e ([HS]), qui comporte le laplacien de Witten en degr\'e 1, ainsi qu'une r\'eduction de Grushin (Feshbach) pour le bas du spectre de cet op\'erateur. La nouveaut\'e est que nous devons maintenant utiliser des probl\`emes de Grushin d'ordre sup\'erieur, ce qui revient \`a \'etudier une plus large fraction du spectre. Dans un cas perturbatif W.M.Wang [W] a r\'ecemment utilis\'e des probl\`emes de Grushin d'ordre sup\'erieur pour obtenir quelques termes dans le d\'eveloppement du taux de d\'ecroissance.\rm \medskip \centerline{\bf 0. Introduction.} \medskip \par In recent years there has been an attempt by B. Helffer, the author and others ([BJS], [H1--4], [HS], [J], [S1--6], [SW], [W]) to apply direct methods to the study of integrals and operators in high dimension, of the type that may appear naturally in statistical mecanics and Euclidean field theory. In the first works, we applied asymptotic methods and noticed already that there is a very strong interplay between asymptotic expansions for integrals obtained by some variant of the stationary phase method and asymptotic solutions of certain Schr\"odinger type operators obtained by the WKB method ([S3--5]). In later works ([S6], [HS], [S4]) we noticed that a suitable version of the maximum principle could be used in the proof of certain asymptotic expansions and to obtain exponential decay of correlations. (In the the work [SW] this was even applied to integrals in the complex domain, and was applied to show exponential decay of the expectation of the Green function for discrete Schr\"odinger operators with random potentials.) \par In the present work we are interested in correlations for Laplace integrals at large distance. In physics language we are interested in the correlations at large distance for continuous spin systems. Under assumptions that imply the exponential decay of these correlations, we want to know the precise rate of exponential decay and to determine the possible polynomial prefactor. The original inspiration came from a talk given by R.Minlos in St Peterburg in 1993 and a corresponding joint paper by him and E.Zhizhina [MZ], about the asymptotics of correlations for discrete spin models at high temperature. Even though we never quite understood the methods used in [MZ], it prompted us to further develop our own methods in the continous spin case and in [S1] we were able to get the leading exponential decay asymptotics for correlations associated to measures of the type $e^{-\phi (x)/h}dx\over \int e^{-\phi (x)/h}dx,$ at large distance, in the semi-classical limit ($h\to 0$). Here $\phi \in C^\infty ({\bf R}^\Lambda ;{\bf R})$, and $\Lambda $ is a finite subset of ${\bf Z}^d$ or a discrete torus of dimension $d$, and we study the limit when $\Lambda $ is large. Recall that the correlation of two functions $u,v$ which do not grow too fast at infinity is given by \ekv{0.1} {{\rm Cor\,}(u,v)=\langle (u-\langle u\rangle )(v-\langle v\rangle ),} where $$\langle u\rangle = {\int u(x)e^{-\phi (x)/h}dx\over \int e^{-\phi (x)/h}dx}$$ denotes the expectation. We also observed that certain associated Schr\"odinger operators, already found with B.Helffer in [HS], are Witten Laplacians in degree 0 or 1. We also managed to replace the use of the maximum principle at many places by $L^2$ methods and consequently we got rid of a certain rigidity in the conditions. The elimination of the maximum principle was not complete however, and the results were flawed by a certain number of unnatural assumptions, in particular that of global uniform strict convexity of the function $\phi $. Another short-coming of [S1] was that we only determined the decay rate and the prefactor up to a factor $(1+{\cal O}(h^{1/2}))$. Moreover, we did not work out the thermodynamical limit ($\Lambda \to{\bf Z}^d$) so oscillations within a factor $1+{\cal O}(h^{1/2})$ could not be excluded, when $\Lambda$ varies. \par With V.Bach and T.Jecko [BJS] we elimintated completely the use of the maximum principle and were able to give simpler and more natural conditions. In particular we could allow the exponent $\phi $ to be strictly convex only near the point where $\phi $ is minimal. The new assumptions still imply that there is only one critical point however. Again we obtained the decay rate and the prefactor only up to a factor $(1+{\cal O}(h^{1/2}))$, and we did not treat the thermodynamical limit. \par The aim of the present paper is to get full asymptotic expansions in powers of $h$ of the decay rate and in powers of the inverse distance and in $h$ of the prefactor, and we shall also treat the thermodynamical limit. To get such more precise results, one has to get higher in the spectrum of the associated Witten Laplacians (or rather something close to that), and we do so by using higher order Grushin problems that we explain more in detail later in this introduction. The main idea of this strategy was rather clear in the author's mind since the writing of [S1], and has become practically realizable with the improvements of [BJS]. W.M.Wang [W] has recently used similar ideas in order to study the rate of exponential decay of correlations when $h=1$ and $\phi $ is a small perturbation of a non-degenerate quadratic form. For the decay rate, she got several terms in an expansion in powers of the perturbation parameter. In principle the method should give full asymptotic expansions and also the prefactor in that case too. [W] also has an interesting application to the exponential decay rate of the Green function for discrete random Schr\"odinger operators. \par We now start to formulate the main result of this paper, and after that we will outline some ideas of the proof. To understand the idea behind many of the assumptions, it may be helpful to have in mind the special case when $\phi =\phi _\Lambda $ is of the form \ekv{0.2} {\phi (x)=\sum_{j\in\Lambda }f(x(j))+\sum_{\vert j-k\vert _1=1} w(x(j),x(k)),} when \ekv{0.3}{\Lambda =({\bf Z}/L{\bf Z})^d,} is a discrete torus with $L\ge 2$ and $\phi $ is of the form \ekv{0.4} {\phi (x)=\sum_{j\in\Lambda }f(x(j))+\sum_{{\vert j-k\vert _1=1}\atop{j\,{\rm or}\, k\in\Lambda }}w(\widetilde{x}(j),\widetilde{x}(k)),} when $\Lambda$ is a finite subset of ${\bf Z}^d$. Here $\vert \cdot \vert _p$ will always denote the $\ell^p$ norm, and when $x$ is the discrete torus $\Lambda $ in (0.3), we let $\vert j\vert _p$ denote $\inf_{\widetilde{j}\in \pi _\Lambda ^{-1}(j)}\vert \widetilde{j}\vert _p$, where $\pi _\Lambda :{\bf Z}^d\to \Lambda $ is the natural projection. In the formula (0.4) we let $\widetilde{x}(j)$ be equal to $x(j)$ for $j\in\Lambda $ and $=0$ otherwise. $f$ and $w$ are smooth real valued functions on ${\bf R}$ and ${\bf R}^2$ respectively, with \ekv{0.5} {w(x,y)=w(y,x).} In the proofs the special forms (0.2), (0.4) will never be of any real advantage, so we shall formulate our main result with more general functions. Notice that if we let $\Lambda $ grow, then the expressions (0.2), (0.4) will in general diverge, while the derivatives of all orders will converge. For that reason (and with the termodynamical limit in mind) we will start by fixing the limiting Hessian, and after that we will see how to get a family of functions $\phi (x)=\phi _\Lambda (x)$ both in the torus case and in the subset case. \par We say that a function $f$ on ${\bf R}^{{\bf Z}^d}$ is smooth if it is continuous for the $\ell^\infty $ topology, differentiable in each of the variables with continuous derivatives (for the $\ell^\infty $ topology) and the derivatives enjoy the same properties et c. Let $\Phi _{j,k}(x)$, $j,k\in{\bf Z}^d$ be smooth and real on ${\bf R}^{{\bf Z}^d}$ and satisfy \ekv{{\rm A}.1} {\Phi _{j,k}(x)=\Phi _{k,j}(x),} \ekv{{\rm A}.2} {\partial _{x_\ell}\Phi _{j,k}=\partial _{x_j}\Phi _{\ell,k},} \ekv{{\rm A}.3} {\Phi =(\Phi _{j,k})\hbox{ is }2\hbox{ standard},} \ekv{{\rm A}.4} {\Phi (0)\ge {\rm Const.}>0.} Here we use the terminology of [S2] concerning $k$ standard tensors. Let $a=(a_{\Lambda ;j,k}(x))$, $x\in{\bf R}^\Lambda $, $j,k\in\Lambda $ be a family of matrices (i.e. 2 tensors) depending on some family of finite sets $\Lambda $. We say that $a=a_\Lambda $ is 2 standard if we have uniformly in $x\in{\bf R}^\Lambda $, $\Lambda $, the estimates \ekv{0.6} {\langle \nabla ^ka(x),t_1\otimes ...\otimes t_{k+2}\rangle ={\cal O}_k(1)\vert t_1\vert _{p_1}..\vert t_k\vert _{p_{k+2}},} for all $t_j\in {\bf C}^\Lambda $ and $p_j\in [1,+\infty ]$ with $1={1\over p_1}+..+{1\over p_{k+2}}$. Here $\vert \cdot \vert _p$ denotes the standard $\ell^p$ norm on ${\bf C}^\Lambda $. When $x$ varies in ${\bf R}^{{\bf Z}^d}$ and $j,k\in{\bf Z}^d$, we require the $a_{j,k}(x)$ to be smooth in the sense mentioned earlier and say that $a$ is 2-standard if (0.6) holds with $\Lambda ={\bf Z}^d$ and $t_j\in{\bf C}^\Lambda $, with $t_j(\lambda )\to 0$, $\Lambda \ni \lambda \to \infty $. Notice that a 2-standard matrix is ${\cal O}(1): \ell^p\to \ell^p$, for $1\le p\le \infty $. \par If we think of the formal expression (0.2) with $\Lambda ={\bf Z}^d$ and assume that $\nabla ^k f$ and $\nabla ^kw$ are bounded for all $k\ge 2$, then the matrix $\Phi (x)=(\phi ''_{j,k}(x))$ fulfills the assumptions (A.1--3). $\Phi (0)$ is the matrix of a convolution and (A.4) amounts to the assumption: \ekv{0.7} {f''(0)+4d\partial _x^2w(0,0)>4d\vert \partial _x\partial _yw(0,0)\vert .} \par If ${\bf Z}^d$ is replaced by a finite set $\Lambda $, then (A.1,2) becomes a necessary and sufficient condition for the existence of a realvalued function $\phi \in C^\infty ({\bf R}^\Lambda )$ with $\phi ''_{j,k}=\Phi _{j,k}$. In the ${\bf Z}^d$ case we shall now see how to produce two different finite dimensional versions of such a function. \par Let $U\subset{\bf Z}^d$ be finite. If $x\in{\bf R}^U$, let $\widetilde{x}\in{\bf R}^{{\bf Z}^d}$ be the zero extension of $x$, so that $\widetilde{x}(j)=x(j)$ for $j\in U$, $\widetilde{x}(j)=0$, for $j\in{\bf Z}^d\setminus U$. Then $$\Phi _{U;j,k}(x):=\Phi _{j,k}(\widetilde{x}),\ j,k\in U$$ is a smooth tensor on ${\bf R}^U$ which satisfies (A.1,2) with $j,k,\ell\in U$. Hence there exists a function $\phi _U(x)\in C^\infty ({\bf R}^U;{\bf R})$ with \ekv{0.8} {\phi ''_{U;j,k}(x)=\Phi _{U;j,k}(x),\ x\in{\bf R}^U,\, j,k\in U.} We make $\phi _U$ unique up to a constant, by requiring that \ekv{0.9} {\phi '_U(0)=0.} It is easy to check that $\phi ''_U$ is 2-standard. \par We next do the same with $U$ replaced by a discrete torus $\Lambda =({\bf Z}/L{\bf Z})^d$. If $\lambda \in {\bf Z}^d$, we define $\tau _\lambda x\in{\bf R}^{{\bf Z}^d}$, by $(\tau _\lambda x)(\nu )=x(\nu -\lambda )$. We will assume translation invariance for $\Phi $: \ekv{{\rm A}.7} {\Phi _{j+\lambda ,k+\lambda }(\tau _\lambda x)=\Phi _{j,k}(x),\ j,k,\lambda \in{\bf Z}^d.} (In section 10 we discuss a larger set of conditions and reproduce here only the most important ones with the same numbering as in section 10.) Notice that if $\Phi _{j,k}$ were the Hessian of a smooth function $\phi \in C^\infty ({\bf R}^{{\bf Z}^d})$ (and the discussion remains valid if we replace ${\bf Z}^d$ by a discrete torus $\Lambda $) then (A.7) would be a consequence of the simpler translation invariance property: \ekv{0.10} {\phi (\tau _\lambda x)=\phi (x).} \par If $x\in{\bf R}^\Lambda $, let $\widetilde{x}=x\circ \pi _\Lambda \in{\bf R}^{{\bf Z}^d}$ be the corresponding $L{\bf Z}^d$ periodic lift, where $\pi _\Lambda :{\bf Z}^d\to\Lambda $ is the natural projection. Replacing $x$ by $\widetilde{x}$ in (A.7), we get \ekv{0.11} {\Phi _{j-\lambda ,k-\lambda }(\widetilde{x})=\Phi _{j,k}(\widetilde{x}),\ \lambda \in L{\bf Z}^d.} If we view $\Phi $ as a matrix, this is equivalent to \ekv{0.12} {\tau _\lambda \circ \Phi (\widetilde{x})=\Phi (\widetilde{x})\circ \tau _\lambda ,\ \lambda \in L{\bf Z}^d,} so $\Phi (\widetilde{x})$ maps $L{\bf Z}^d$ periodic vectors into the same kind of vectors. Hence there is a naturally defined $\Lambda \times \Lambda $ matrix $\Phi _\Lambda (x)$, defined by \ekv{0.13} { \widetilde{\Phi _\Lambda (x)t}=\Phi (\widetilde{x})\widetilde{t}, } where again the tilde indicates that we take the periodic lift. On the matrix level, we get \ekv{0.14} { \Phi _{\Lambda ;j,k}(x)=\sum_{\widetilde{k}\in\pi _\Lambda ^{-1}(k)}\Phi _{\widetilde{j},\widetilde{k}}( \widetilde{x}), } for any $\widetilde{j}\in \pi _\Lambda ^{-1}(j)$. Alternatively, we have \ekv{0.15} {\Phi _{\Lambda ;j,k}(x)=\sum_{\widetilde{j}\in\pi _\Lambda ^{-1}(j)}\Phi _{\widetilde{j},\widetilde{k}}( \widetilde{x}),\ \widetilde{k}\in \pi _\Lambda ^{-1}(k),} and $\Phi _{\Lambda ;j,k}$ is symmetric (cf. (A.1)). \par In section 10, we shall verify that $\Phi _\Lambda $ satisfies (A.1--4), so there exists $\phi _\Lambda \in C^\infty ({\bf R}^\Lambda ;{\bf R})$, unique up to a constant, such that \ekv{0.16} {\Phi _{\Lambda ;j,k}(x)=\partial _{x_j}\partial _{x_k}\phi _\Lambda (x),\ \phi _\Lambda '(0)=0.} \par In the case when $\Phi $ is the Hessian of the formal expression (0.2) with $\Lambda $ there replaced by ${\bf Z}^d$, and if we assume that 0 and (0,0) are critical points of $f$ and $w$ respectively, then it is easy to see that $\phi _\Lambda $ is given by the expression (0.2) when $\Lambda $ is a discrete torus with $L\ge 3$ and by (0.4) when $\Lambda =U$ is a bounded subset of ${\bf Z}^d$. \par We assume that $\Phi (0)$ is ferromagnetic in the sense that \ekv{{\rm A}.9}{\Phi _{j,k}(0)\le 0,\ j\ne k.} We have \ekv{0.17} {\Phi (0)= 1-\widetilde{v}_0*, } where $0\le \widetilde{v}_0\in\ell^1({\bf Z}^d)$ is even with $\widetilde{v}_0(0)=0$ and the star indicates that $\widetilde{v}_0$ acts as a convolution. Actually, the constant 1 should be replaced by a more general constant $a>0$, but we may always reduce ourselves to the case $a=1$, by a dilation in $h$. \par Assume that there exists a finite set $K\subset {\bf Z}^d$ such that \ekv{{\rm A}.10} {\widetilde{v}_0(j)>0,\, j\in K,\ {\rm Gr\,}(K)={\bf Z}^d,} where ${\rm Gr\,}(K)$ denotes the smallest subgroup of ${\bf Z}^d$ which contains $K$. We also make the following finite range assumption: \ekv{{\rm A.fr}} {\exists C_0,\hbox{ such that }\Phi _{j,k}(x)=0\hbox{ for }\vert j-k\vert >C_0.} \par We introduce the $2$ standard matrix \ekv{0.18} {A(x)=\int_0^1 \Phi (tx) dt,} The following assumption is a weakened convexity assumption and will be used in section 10 together with a maximum principle (from [S4]) to obtain other more explicit conditions. \eekv{{\rm A.mp}} {\exists \epsilon _0>0\hbox{ such that for every $x\in {\bf R}^{{\bf Z}^d}$, $A(x)$ satisfies (${\rm mp\,}\epsilon _0$): If }} {\hbox{$t\in \ell^1({\bf Z}^d)$, $s\in \ell^\infty ({\bf Z}^d)$, and $\langle t,s\rangle =\vert t\vert _1\vert s\vert _\infty $, then $\langle A(x)t,s\rangle \ge \epsilon _0 \vert t\vert _1\vert s\vert _\infty $.}} Notice that this assumption is fulfilled if $A(x)=1+B(x)$ with $\Vert B(x)\Vert _{{\cal L}(\ell^\infty ,\ell^\infty )}\le 1-\epsilon _0$. Also notice that (A.4) is a consequence of (A.mp). \par Let $U_j\in {\bf Z}^d$, $j=1,2,..$ be an increasing sequence of finite sets containing $0$ and converging to ${\bf Z}^d$. Let $2\le L_j\nearrow\infty $ be a sequence of integers with \ekv{0.19} {U_j\subset [-{L_j\over 4},{L_j\over 4}]^d,} and let $\Lambda =\Lambda _j=({\bf Z}/L_j{\bf Z})^d$ be a corresponding sequence of discrete tori, so that we can view $U_j$ as a subset of $\Lambda _j$ in the natural way. \par The following is the main result of our work: \medskip \par\noindent \bf Theorem 0.1. \it Let $\Phi _{j,k}(x)\in C^\infty ({\bf R}^{{\bf Z}^d})$ satisfy (A.1--3, 7, 9, 10, fr, mp) and define $\phi _U(x)\in C^\infty ({\bf R}^U;{\bf R})$, $\phi _\Lambda \in C^\infty ({\bf R}^\Lambda ;{\bf R})$ as above, when $U\subset {\bf Z}^d$ is finite and $\Lambda =({\bf Z}/L{\bf Z})^d$ is a discrete torus. Let $U_j$, $\Lambda _j$ be as above, and put $r_j:={\rm dist\,}(0,{\bf Z}^d\setminus U_j)$, so that $r_j\nearrow +\infty $ when $j\to\infty $. \par Then there exist $C_0\ge 1$, $j_0\in{\bf N}$, $\theta >0$, $h_0>0$, such that for $j\ge j_0$, $00$ uniformly in $j$. If we had assumed (as in [S1]) that $\phi ''(x)\ge {\rm const.}>0$ uniformly in $x,j$, that would have been immediate from (0.31). As in [BJS] we only assume this at $x=0$ however, and the idea (exploited in [BJS]) is then to make a limited Taylor expansion, $$\phi ''(x)=\phi ''(0)+\sum_\nu A_\nu (x)\phi _{x_\nu }'(x),$$ to write $\phi '_{x_\nu }(x)$ as $h^{1\over 2}(Z_\nu +Z_\nu ^*)$, and to use a priori estimates that give control over $\Vert Z_\nu u\Vert $. \par In [HS], we established a general formula for the correlations and in [S1] we observed that it is related to Witten Laplacians. In this formalism and under the normalization condition (0.25) it reads: \ekv{0.33} {{\rm Cor\,}(u,v)=({\Delta _\phi ^{(1)}}^{-1}d_\phi (e^{-\phi /2h}u)\vert d_\phi (e^{-\phi /2h}v))=h({\Delta _\phi ^{(1)}}^{-1}(e^{-\phi /2h}du)\vert (e^{-\phi /2h}dv)).} In [HS], we used such a formula to establish the exponential decay of the correlations ${\rm Cor\,}(x_\nu ,x_\mu )$ when ${\rm dist\,}(\nu ,\mu )$ is large. This is based on the simple idea that since we have a uniform bound on the norm of $(\Delta _\phi ^{(1)})^{-1}$, then we should also have such a bound after a conjugation of this operator by an exponential weight $\rho (\nu )=e^{r(\nu )}$, $\nu \in\Lambda $, provided that $r$ does not vary too fast. \par In [S1] we obtained the leading behaviour of ${\rm Cor\,}(x_\nu ,x_\mu )$ for large ${\rm dist\,}(\nu ,\mu )$ by using a Feshbach (or Grushin) approach to $\Delta _\phi ^{(1)}$ which in many ways amounts to study the bottom of the spectrum of this operator. We introduced the auxiliary operator $R_+= R_+^{1,0}:L^2({\bf R}^\Lambda )\to\ell^2(\Lambda )$ by \ekv{0.34}{(R_+^{1,0}u)(j)=(u\vert e^{-\phi /2h}dx_j)=(u_j\vert e^{-\phi /2h}),\ j\in\Lambda ,} where $u=\sum u_jdx_j\simeq (u_j)_{j\in\Lambda }$, so in each component, we project onto the kernel of $\Delta _\phi ^{(0)}$. Let $R_-^{1,0}=(R_+^{1,0})^*$ be the adjoint. \par Let $${\cal H}_1=\{ u\in L^2({\bf R}^\Lambda );\, Z_\nu u\in L^2,\forall \nu \in\Lambda \}$$ with the corresponding norm $$\Vert u\Vert _{{\cal H}_1}^2=\Vert u\Vert ^2+\sum_\nu \Vert Z_\nu \Vert ^2,$$ where $\Vert \cdot \Vert $ denotes the $L^2$ norm. Let ${\cal H}_{-1}={\cal H}_1^*$ denote the dual space. Then as we shall prove below (and as was essentially proved in [S1] and in greater generality in [BJS]), the operator \eekv{0.35} { {\cal P}^{0,1}(z)=\pmatrix{\Delta _\phi ^{(1)}-z &R_-^{0,1}\cr R_+^{0,1}&0}:(\ell^2(\Lambda )\otimes {\cal H}_1)\times (\ell^2(\Lambda )\otimes {\bf C})\to} {\hskip 6cm (\ell^2(\Lambda )\otimes {\cal H}_{-1})\times (\ell^2(\Lambda )\otimes {\bf C}) } is bijective with a uniformly bounded inverse \ekv{0.36} {{\cal E}^{0,1}(z)=\pmatrix{E^{0,1}(z)&E_+^{0,1}(z) \cr E_-^{0,1}(z) & E_{-+}^{0,1}(z) }} for \ekv{0.37} {-C\le z\le 2\lambda _{{\rm min}}(\phi ''(0))-{1\over C},} when $h$ is small enough depending on $C$, and $C\ge 1$ may be arbitrary. \it Here and in the following we follow the convention that all estimates and assumptions will be uniform w.r.t. $\Lambda $, if nothing else is specified. \rm By $\lambda _{{\rm min}}(\phi ''(0))$, we denote the lowest eigenvalue of $\phi ''(0)$. (As a matter of fact, we will need the invertibility only for $z=0$ but keeping track of the spectral parameter will help the understanding. In the end classes of exponential weights will be the more appropriate objects.) \par Further, as we shall see (and as was established in [S1], [BJS]), we have \eekv{0.38} {E_+^{0,1}=R_-^{0,1}+{\cal O}(h^{1\over 2}),\ \ E_-^{0,1}=R_+^{0,1}+{\cal O}(h^{1\over 2}),} {E_{-+}^{0,1}=z-\phi ''(0)+{\cal O}(h^{1\over 2}),} in the respective spaces of bounded operators. Notice that $E_{-+}^{0,1}(z)$ is invertible for $-C\le z\le \lambda _{{\rm min}}(\phi ''(0))-{1\over C}$, i.e. in a smaller domain than (0.37). Actually, instead of varying the spectral parameter, we shall take $z=0$ and conjugate ${\cal P}^{0,1}(0)$ by an exponential weight$\pmatrix{\rho \otimes 1&0\cr 0& \rho \otimes 1}$, with $\rho =e^r:\Lambda \to ]0,\infty [$. We shall then see that the conjugated operator ${\cal P}^{0,1}$ is uniformly invertible for $\rho $ in a large class of weights. Notice that the inverse is simply $$\pmatrix{\rho \otimes 1 &0\cr 0& \rho \otimes 1}{\cal E}^{0,1}(0)\pmatrix{\rho ^{-1}\otimes 1 &0\cr 0 & \rho ^{-1}\otimes 1}.$$ Moreover we shall see that (0.38) remains valid for the conjugated operators. $(E_{-+}^{0,1})^{-1}$ will cope with conjugation only with weights in a smaller class, and starting with the case when $\Lambda$ is a discrete torus (implying that $E_{-+}^{0,1}$ is a convolution), we shall be able to analyze quite precisely the rate of decay of this inverse, and see that it corresponds to weights in the larger class of weights with which ${\cal E}^{1,0}$ accomodates conjugation. Since $$(\Delta _\phi ^{(1)})^{-1}=E^{0,1}(0)-E_+^{0,1}(0)(E_{-+} (0))^{-1}E_-^{0,1}(0),$$ we can apply (0.33) and get \eekv{0.39} {{\rm Cor\,}(x_\nu ,x_\mu )=h(E^{0,1}(0)(e^{-\phi /2h}dx_\nu )\vert (e^{-\phi /2h}dx_\mu ))-} {\hskip 4cm h((E_{-+}(0))^{-1}E_-^{0,1}(0)(e^{-\phi /2h}dx_\nu )\vert E_-^{0,1}(0)(e^{-\phi /2h}dx_\mu )).} Because ${\cal E}^{0,1}$ can cope with conjugation with stronger exponential weights than $(E_{-+}^{0,1})^{-1}$, we see that the first term of the RHS in (0.39) has faster decay, than the second, when ${\rm dist\,}(\nu ,\mu )\to\infty $ and the more precise information evocated about the inverse of $E_{-+}$ together with (0.38) leads to a result of the type (0.20), where a priori the $p_{1,h}$ and $q$ will depend also on $\Lambda $ through factors $1+{\cal O}(h^{1/2})$. So far the ideas were already developed in [S1] and [BJS]. \par In order to get complete expansions as stated in the theorem, we will introduce higher order Grushin problems. Let ${\bf N}_j^\Lambda $ be the set of multiindices $\alpha :\Lambda \to {\bf N}$ of length $j$: $\vert \alpha \vert =\vert \alpha \vert _1=j$. If $J$ is a finite subset of ${\bf N}$, we put ${\bf N}_J=\cup_{j\in J}{\bf N}_j^\Lambda $. Since the $Z_\nu ^*$ form a commutative family, the operator ${1\over \alpha !}(Z^*)^\alpha $ is well-defined. For $u\in L^2({\bf R}^\Lambda )$, put $$(R_+^{N,0}u)(\alpha )=(u\vert {1\over\alpha !}(Z^*)^\alpha (e^{-\phi /2h})),\ \vert \alpha \vert \le N,$$ so that $R_+^{N,0}:L^2({\bf R}^\Lambda )\to\ell ^2({\bf N}^\Lambda _{[0,N]})$. (We will see in section 4 that this operator is uniformly bounded.) Notice that ${1\over\alpha !}(Z^*)^\alpha (e^{-\phi /2h})$ are Hermite functions when $\phi $ is a quadratic form. When $u\in \ell^2(\Lambda )\otimes L^2({\bf R}^\Lambda )$, we put $(R_+^{N,1}u)(j,\alpha )=(R_+^{N,0}u_j)(\alpha )$, $(j,\alpha )\in \Lambda \times {\bf N}_{[0,N]}^\Lambda $. Let $R_-^{N,k}=(R_+^{N,k})^*$, $k=0,1$, and introduce the auxiliary (Grushin) operators for $k=0,1$: \eekv{0.40} {{\cal P}^{N,k}(z)=\pmatrix{ \Delta _\phi ^{(k)} &R_+^{N,k}\cr R_-^{N,k} &0}:} {\cases{ {\cal H}_1\times \ell^2({\bf N}_{[0,N]}^\Lambda )\to{\cal H}_{-1}\times \ell^2({\bf N}_{[0,N]}^\Lambda ),\ k=0, \cr (\ell^2(\Lambda )\otimes {\cal H}_1)\times (\ell^2(\Lambda )\otimes \ell^2({\bf N}_{[0,N]}^\Lambda ))\to(\ell^2(\Lambda )\otimes {\cal H}_{-1})\times (\ell^2(\Lambda )\otimes \ell^2({\bf N}_{[0,N]}^\Lambda )),\ k=1.}} We will see in section 5 that ${\cal P}^{N,0}(z)$ is uniformly invertible for\hfill\break $-C\le z\le (N+1)\lambda _{{\rm min}}(\phi ''(0))-{1\over C}$ and that the same is true for ${\cal P}^{N,1}(z)$ in the range $-C\le z\le (N+2)\lambda _{{\rm min}}(\phi ''(0))-{1\over C}$. This will be proved following the inductice scheme $$(N,1)\to (N+1,0)\to (N+1,1),$$ starting with the case $(-1,1)$, where by definition ${\cal P}^{-1,1}(z)=\Delta _\phi ^{(1)}-z$. \par If $${\cal E}^{N,k}(z)=\pmatrix{E^{N,k}(z) &E_+^{N,k}(z)\cr E_-^{N,k}(z) &E_{-+}^{N,k}(z)}$$ denotes the inverse of ${\cal P}^{N,k}(z)$ and $E^{N,k}_{-+;\nu ,\mu }(z)$ denotes the operator matrix element of $E_{-+}^{N,0}$ corresponding to the decomposition $\ell^2({\bf N}_{[0,N]}^\Lambda )=\oplus_{\nu =0}^N\ell^2({\bf N}_\nu ^\Lambda )$, we will further see that \ekv{0.41} {E_{-+;\nu ,\mu }^{N,0}(z)=h^{{1\over 2}\vert \nu -\mu\vert }B^N_{\nu ,\mu }(z;h)+{\cal O}(h^{{1\over 2}(\vert N+1-\nu \vert +\vert N+1-\mu \vert )})\hbox{ in }{\cal L}(\ell^2,\ell^2),} where $B^N_{\nu ,\mu }$ has a complete asymptotic expansion in powers $h^\ell$, $\ell\in{\bf N}$. Essentially the same result holds for $E_{-+}^{N,1}$ and similar results hold for $E_{\pm}^{N,k}$. The idea behind this result is to consider the matrix of $\Delta _\phi ^{(0)}$ (and similarly for $\Delta _\phi ^{(1)}$) with respect to the decomposition $$L^2({\bf R}^\Lambda )={\cal L}_0\oplus..\oplus{\cal L}_N\oplus{\cal L}_{[0,N]}^\perp ,$$ where ${\cal L}_j=R_-^{N,0}(\ell^2({\bf N}_j^\Lambda ))$ and ${\cal L}_{[0,N]}^\perp$ is the orthogonal of ${\cal L}_0\oplus ..\oplus{\cal L}_N={\cal L}_{[0,N]}$, for which the corresponding matrix elements of $(\Delta _\phi ^{(0)})_{\nu ,\mu }$ should behave as in (0.41). Notice that (0.41) gives increasing precision in the asymptotics for a fixed $(\mu ,\nu )$, when $N$ increases. It is possible to describe ${\cal E}^{M,k}$ in terms of ${\cal E}^{N,k}$, for $M\le N$, and using this with $M=0$ and $N\to \infty $, we arrive at a complete asymptotic expansion of $E_{-+}^{0,1}(z;h)$ and at similar almost complete descriptions of $E_{\pm}^{0,1}(z;h)$. In other words, by using higher order Grushin problems it is possible to improve (0.38) and to get full asymptotics. This improvement also survives the conjugation by exponential weights in a sufficiently large class, and leads to a complete asymptotic description of $(E_{-+}^{0,1}(z))^{-1}$, including the decay rate at large distances. Finally we use this improved information in (0.39) to get complete asymptotics of the correlations. The handling of the thermodynamical limit requires some additional arguments that we do not discuss here. \par A major motivation for this paper was the hope (yet to be fulfilled) that the use of higher order Grushin problems may be useful in the study of correlations in cases when $\phi $ is only weakly convex at its critical point. In such cases we do not always expect the correlations to decay exponentially any more and one may expect phenomena like phonons in crystals. Though we are still far from such a result, we may point out that the parameter $N$ can be interpreted as a maximum number of particles under consideration, and that the $k$ particle space $\ell^2({\bf N}_k^\Lambda )$ can be identified with the $k$ fold symmetric tensor product of $\ell^2(\Lambda )$ with itself. In other words, our particles are bosons. \par In sections 1--9, we do all the essential work, adding successively the assumptions that we need. At the end of section 9, we arrive at the main result. In section 10, we consider a slightly less general framework and extract a main result which is more easily formulated. \par\noindent In section 1 we review some standard facts about Witten Laplacians. \par\noindent In section 2 we introduce some special Sobolev spaces, which are the natural ones for our variational point of view. \par\noindent In section 3 we discuss how to reshuffle creation and annihilation operators. The reason for doing so will appear very naturally, and we are aware of the fact that such reorderings also appear in quantum field theory. \par\noindent In section 4 we apply the result of the preceding section to study certain scalar producs. \par\noindent Section 5 is devoted to the well-posedness of higher order Grushin problems. \par\noindent In section 6 we get asymptotics for the solutions of these problems and in section 7, we show that these asymptotics for ${\cal P}^{N,1}$ remain after introducing certain exponential weights on the $\ell^2(\Lambda )$ component of ${\cal P }^{N,1}$. \par\noindent In section 8, we study the effect of parameter dependence in order to treat the thermodynamical limit. \par\noindent In section 9 we arrive at the main result on the asymptotics of the correlations also in the thermodynamical limit. \par\noindent In section 10 we extract the main result as it is formulated in Theorem 0.1 above. \par\noindent The two appendices can be read when referred to in the main text. \medskip \par\noindent \bf Acknowledgements. \rm This work was supported by the TMR--network FMRX-CT 96-0001 "PDE and QM". We have benefitted from interesting discussions with W.M.Wang. %\vfill\eject \bigskip \centerline{\bf 1. Assumptions on $\phi $.} \medskip \par Let $\phi \in C^\infty ({\bf R}^\Lambda ;{\bf R})$, where $\Lambda $ is a finite set. We shall let $\Lambda $ and consequently $\phi $ vary with some parameter, but all assumptions are uniform w.r.t. $\Lambda $, if nothing else is specified. Our first assumption is \eeekv{{\rm H}1} {\phi ^{(2)}=\phi ''\hbox{ is 2 standard in the sense that for every $k\ge 2$, we have uniformly} } {\hbox{in $\Lambda $ and in $x\in{\bf R}^\Lambda $}: \langle \phi ^{(k)}(x),t_1\otimes\dots\otimes t_k\rangle ={\cal O}(1)\vert t_1\vert _{p_1}\dots \vert t_k\vert _{p_k},\ t_j\in{\bf C}^\Lambda ,} {\hbox{whenever }1\le p_j\le \infty ,\hbox{ and }1={1\over p_1}+\dots +{1\over p_k}.} Here $\phi ^{(k)}=\nabla ^k\phi $ is the symmetric tensor of $k$th order derivatives. See [S2] for definitions and basic properties concerning standard tensors. Recall that by complex interpolation it suffices to have the estimate in (H1) in the extreme cases $$p_\nu =\cases{1,\ \nu =j,\cr \infty ,\ \nu \ne j,}$$ for $j=1,..,k$. Notice that (H1) implies that $\phi ''(x):\ell^p\to\ell^p$ is uniformly bounded for $x\in{\bf R}^\Lambda $, $1\le p\le \infty $. \par The next three assumptions imply that $x=0$ is a non-degenerate critical point of $\phi $ and the only critical point: \ekv{{\rm H}2}{\phi '(0)=0,} \ekv{{\rm H}3}{\phi ''(0)\ge {\rm const.}>0,} \eekv{{\rm H}4} {\phi '(x)=A(x)x\hbox{ where $A(x)$ is 2 standard and has an}} {\hbox{inverse }B(x)\hbox{ which is }{\cal O}(1):\ell^p\to\ell^p,\ 1\le p\le \infty .} We observe that $B$ will also be 2 standard. Also notice that (H1), (H2) imply that \ekv{1.1} {\vert \phi '(x)\vert _p\le {\cal O}(1)\vert x\vert _p,\ 1\le p\le \infty ,} while (H4) implies the reverse estimate \ekv{1.2}{\vert x\vert _p\le{\cal O}(1)\vert \phi '(x)\vert _p,\ 1\le p\le \infty .} It would be of interest to know if conversely (1.2) and (H1--3) imply (H4). Also notice that (H4) (or (1.2)) implies that $\phi ''(0)^{-1}$ exists and is ${\cal O}(1):\ell^p\to\ell^p$. When checking (H4), a natural candidate for $A(x)$ is $\int_0^1\phi ''(tx)dt$, which is 2 standard by (H1). \par We end this section by introducing Witten Laplacians and related objects (cf. [S]). For that purpose we shall work on ${\bf R}^\Lambda $, where $\Lambda $ is some finite set. Let $d=\sum_{\ell\in \Lambda }dx_\ell^\wedge\otimes \partial _{x_\ell}$ denote the DeRahm exterior differentiation which takes differential $k$ forms on ${\bf R}^\Lambda $ to differential $k+1$ forms. Here $dx_\ell^\wedge$ denotes the operator of left exterior multiplication by $dx_\ell$ and we let $dx_\ell^\rfloor$ denote the adjoint operator of contraction, which is well defined if we view ${\bf R}^\Lambda $ as a Riemannian manifold with the standard metric. Recall that $d$ is a complex in the sense that $d\circ d=0$. Using the standard scalar product on the space of smooth $k$ forms, we can define the adjoint $d^*=\sum_{\ell\in\Lambda }dx_\ell^\rfloor\otimes (-\partial _{x_\ell})$, taking $k+1$ forms into $k$ forms. The corresponding Hodge Laplacian is then $d^*d+dd^*$. It conserves $k$ forms and commutes with $d$ and $d^*$. \par The Witten exterior differentiation is obtained from $d$ by conjugation by $e^{\phi /h}$ and multiplication by a cosmetic factor: \ekv{1.3} {d_\phi :=h^{1/2}e^{-\phi /2h}\circ d\circ e^{\phi /2h}=\sum_{\ell\in\Lambda }dx_\ell^\wedge\otimes Z_\ell,} where \ekv{1.4}{Z_\ell=e^{-\phi /2h}\circ h^{1/2}\partial _{x_\ell}\circ e^{\phi /2h}=h^{1/2}\partial _{x_\ell}+h^{-1/2}\partial _{x_\ell}\phi /2.} We view $Z_\ell$ as annihilation operators. The corresponding creation operators are \ekv{1.5}{Z_\ell^*=-h^{1/2}\partial _{x_\ell}+h^{-1/2}\partial _{x_\ell}\phi/2 .} We have the commutation relations: \ekv{1.6}{[Z_j,Z_k]=0,\ [Z_j,Z_k^*]=\phi _{j,k}''(x),\ j,k\in \Lambda .} \par $d_\phi $ is a complex and the corresponding Hodge Laplacian is called the Witten Laplacian and is given by: \ekv{1.7}{\Delta _\phi =d_\phi ^*d_\phi +d_\phi d_\phi ^*.} It conserves the degree of forms and we denote by $\Delta _\phi ^{(k)}$ the restriction to $k$ forms. Only the cases $k=0,1$ will be of importance to us and maybe the explanation of this fact is that by working with differential forms, we impose a fermionic structure, while the problems in this paper have a bosonic structure with the degree $k$ viewed as the number of particles. It would be interesting to know if there are some other operators better adapted to the bosonic structure. A general formula for $\Delta _\phi $ is \ekv{1.8}{\Delta _\phi =I\otimes\Delta _\phi ^0+\sum_{j,k}\phi _{j,k}''(x)dx_j^\wedge\,dx_k^\rfloor ,} where we from now adopt the convention of letting the form component be the first factors and the function components to be the last factors when we represent differential forms and corresponding operators as tensor products. $\Delta _\phi ^{(0)}$ acts on scalar functions and is given by: \ekv{1.9} {\Delta _\phi ^{(0)}=\sum_j Z_j^*Z_j.} When $k=1$, the formula (1.8) simplifies to \ekv{1.10} {\Delta _\phi ^{(1)}=I\otimes \Delta _\phi ^{(0)}+\phi ''(x),} provided that we view $1$ forms as functions with values in ${\bf C}^\Lambda $. Again $d_\phi $ and $d_\phi ^*$ commute with $\Delta _\phi $ and in particular, \ekv{1.11}{d_\phi \Delta _\phi ^{(0)}=\Delta _\phi ^{(1)}d _\phi ,\ d_\phi^* \Delta _\phi ^{(1)}=\Delta _\phi ^{(0)}d _\phi^*.} \par Under the assumptions (H1-4) we know (see for instance [BJS] or [Jo]) that $\Delta _\phi ^{(k)}$ can be realized as a selfadjoint operator by means of the Friedrichs extension. We will use the same symbol to denote this selfadjoint operator. Moreover, the spectrum is discrete and contained in $[0,+\infty [$. When $k=0$ the lowest eigenvalue is simple and equal to $0$. The corresponding eigenspace is spanned by $e^{-\phi /2h}$. When $k=1$, the lowest eigenvalue is $>0$ (see for instance [S]). %\vfill\eject \bigskip \centerline{\bf 2. The spaces ${\cal H}_{\pm 1}$.} \medskip \par There will be two versions of these spaces, one for scalar (${\bf C}$ valued) functions and one for functions with values in ${\bf C}^\Lambda $. We start with the scalar case. We assume (H1--4) throughout this section. \par If $u\in C_0^\infty ({\bf R}^\Lambda )$, we put \ekv{2.1}{\Vert u\Vert _{{\cal H}_1}^2=\Vert u\Vert _1^2=\Vert u\Vert ^2+\sum_{\ell\in\Lambda }\Vert Z_\ell u\Vert ^2,} and let ${\cal H}_1$ be the closure of $C_0^\infty $ for this norm. ${\cal H}_1$ is the form domain of $\Delta _\phi ^{(0)}$, and by a standard regularization argument we know that \ekv{2.2}{{\cal H}_1=\{ u\in L^2({\bf R}^\Lambda );\, Z_\ell u\in L^2,\,\forall \ell\in\Lambda \}.} Let ${\cal H}_{-1}$ be the dual space. Using the standard $L^2$ inner product, we view ${\cal H}_{-1}$ as a space of temperate distributions and have the natural inclusions: \ekv{2.3} {{\cal S}({\bf R}^\Lambda )\subset{\cal H}_1\subset{\cal H}_0\subset{\cal H}_{-1}\subset{\cal S}'({\bf R}^\Lambda ).} Here the two inclusions in the middle correspond to inclusion operators of norm $\le 1$ and ${\cal H}_0$ denotes $L^2({\bf R}^\Lambda )$. ${\cal H}_1$ is a Hilbert space with scalar product \ekv{2.4}{[u\vert v]_1=(u\vert v)+\sum_{\ell\in \Lambda }(Z_\ell u\vert Z_\ell v)=((1+\Delta _\phi ^{(0)})u\vert v),} where $(\cdot \vert \cdot \cdot )$ is the usual innerproduct in $L^2$. From this it follows that $1+\Delta _\phi ^{(0)}$ is unitary from ${\cal H}_1$ to ${\cal H}_{-1}$. We also remark that ${\cal H}_{-1}$ is the space of all \ekv{2.5}{u=u^0+\sum Z_\ell^*u_\ell ,} with $u^0,u_\ell\in L^2$. Moreover $\Vert u\Vert _{-1}^2$ is the infimum of $\Vert u^0\Vert ^2+\sum \Vert u_\ell\Vert ^2$ over all decompositions as in (2.5). \par We now pass to spaces of 1 forms, whenever there is a possibility of confusion we indicate the degree of the forms by a superscript $(k)$, so that the spaces just defined are ${\cal H}_{\pm 1}^{(0)}$. Put \ekv{2.6} {{\cal H}_{\pm 1}^{(1)}=\ell^2(\Lambda )\otimes {\cal H}_{\pm 1}^{(0)}.} The corresponding scalar product of two 1 forms $u=\sum u_jdx_j$, $v=\sum v_jdx_j$ is then \ekv{2.7} {[u\vert v]_1=\sum [u_j\vert v_j]_1=\sum_j (u_j\vert v_j)+\sum_{j,k}(Z_ju_k\vert Z_jv_k)=(u\vert v)+\sum_{j,k}(Z_ju_k\vert Z_jv_k).} It can also be written $((1+1\otimes\Delta _\phi ^{(0)})u\vert v)$. Again ${\cal H}_{-1}^{(1)}$ is the dual space of ${\cal H}_{1}^{(1)}$, and \ekv{2.8}{(1+1\otimes\Delta _\phi ^{(0)}):{\cal H}_1^{(1)}\to{\cal H}_{-1}^{(1)}\hbox{ is unitary.}} \par Later we will need to approximate $\phi ''(x)$ by $\phi ''(0)$ in these spaces, and for that we shall use the following lemma. \medskip \par\noindent \bf Lemma 2.1. \it The operator $u(x)\mapsto (\phi ''(x)-\phi ''(0))u(x)$ is bounded ${\cal H}_1^{(1)}\to {\cal H}_{-1}^{(1)}$ and of norm ${\cal O}(h^{1/2})$. \medskip \par\noindent \bf Proof. \rm Using Proposition A.1, we see that \ekv{2.9} {\phi_{j,k} ''(x)-\phi_{j,k} ''(0)=h\phi ^{(1)}_{j,k}(x)+h^{1/2}\sum_{\ell}Z_\ell^*\circ \phi ^{(0)}_{j,k,\ell}(x)+h^{1/2} \sum_{\ell} \phi ^{(0)}_{j,k,\ell}(x)\circ Z_\ell,} where $\phi ^{(\nu )}$ are standard tensors. Let $u,v\in C_0^\infty ({\bf R}^\Lambda )$ and use (2.9) to get \eekv{2.10} {((\phi ''(x)-\phi ''(0))u\vert v)}{\hskip 1cm =h^{1/2}\sum_{j,k,\ell}( \phi _{j,k,\ell}^{(0)}u_k\vert Z_\ell v_j)+h^{1/2}\sum_{j,k,\ell}(\phi _{j,k,\ell}^{(0)}Z_\ell u_k\vert v_j)+h\sum_{j,k}(\phi ^{(1)}_{j,k}u_k\vert v_j).} Since $\phi ^{(1)}$ is 2 standard, the last sum is ${\cal O}(h)\Vert u\Vert \Vert v\Vert $. For the two other sums, we use Lemma B.2 and get for the first sum: \ekv{2.11} {\vert \sum_{\ell ,j,k} \phi _{j,k,\ell}^{(0)}u_k \overline{Z_\ell v_j}\vert \le {\cal O}(1)(\sum_k \vert u_k(x)\vert ^2)^{1/2}(\sum _{j,\ell}\vert Z_\ell v_j\vert ^2)^{1/2}.} This implies that the first sum in (2.10) is ${\cal O}(1)\Vert u\Vert \Vert v\Vert _1$. Similarly the second sum is ${\cal O}(1)\Vert u\Vert _1\Vert v\Vert $. We then get \ekv{2.12} {((\phi ''(x)-\phi ''(0))u\vert v)={\cal O}(h^{1/2})\Vert u\Vert _1\Vert v\Vert _1,} which implies the lemma.\hfill{$\#$} %\vfill\eject \bigskip \centerline{\bf 3. Reshuffling of $Z$ and $Z^*$.} \medskip \par Let $J=\{ 1,..,N\}$, $K=\{ 1,..,M\}$, $f\in C^\infty ({\bf R}^\Lambda )$. Then for $j\in \Lambda ^J$, $k\in \Lambda ^K$, we want to rewrite \ekv{3.1}{(\prod_{\nu \in J}Z_{j(\nu )})\circ f\circ (\prod_{\mu \in K}Z_{k(\mu )}^*)} as a sum of similar terms with the $Z^*$ to the left and the $Z$ to the right. We first move each factor $Z_{j(\nu )}$ as far as possible to the right, taking into account the appearance of commutator terms, due to the relations \ekv{3.2} {[Z_j,f]=h^{1\over 2}\partial _{x_j}f(x)=[f,Z_j^*],\ [Z_j,Z_k^*]=\phi _{j,k}''(x).} After that, we move the surviving factors $Z_k^*$ as a far as possible to the left, generating new commutator terms. The expression (3.1) becomes \eekv{3.3} {\hskip -1cm\sum_{P\ge 0}{1\over P!}\sum_{{{J=J_0\cup ..\cup J_{P+1}}\atop{K=K_0\cup ..\cup K_{P+1}}}\atop{{{\rm partitions\ with}}\atop {J_p\ne\emptyset\ne K_p},\ {\rm for}\ 1\le p\le P}}(\prod_{\mu \in K_0}Z_{k(\mu )}^*)\circ h^{{1\over 2}(\# J_{P+1}+\# K_{P+1})}((\prod_{\mu \in K_{P+1},\atop \nu \in J_{P+1}}\partial _{x_{k(\mu )}}\partial _{x_{j(\nu )}})f)\times } {\prod_{p=1}^P(h^{-1+{1\over 2}(\# J_p+\# K_p)}(\prod_{\mu \in K_p\atop \nu \in J_p}(\partial _{x_{k(\mu )}}\partial _{x_{j(\nu )}}))\phi )\prod_{\nu \in J_0}Z_{j(\nu )}. } Here and in the following we use the term partition for a union of pairwise disjoint sets. The factor ${1\over P!}$ can be eliminated if we let the second summation be over all similtaneous partitions of $J$ and $K$ which are non-ordered in the indices $1\le p\le P$. \par Define a map $m: \Lambda ^N\to {\bf N}^\Lambda $, by \ekv{3.4} {m(j)(\lambda )=\# \{k;\, j(k)=\lambda \},\ \lambda \in\Lambda . } If $\alpha\in {\bf N}^\Lambda $, we put $\vert \alpha \vert =\vert \alpha \vert _1=\sum_{\lambda \in\Lambda }\alpha (\lambda )$. Then $\vert m(j)\vert =N$. We write \ekv{3.5} { {\bf N}_N^\Lambda =\{ \alpha \in{\bf N}^\Lambda ; \vert \alpha \vert =N\}, } and more generally \ekv{3.6} { {\bf N}_A^\Lambda =\{ \alpha \in{\bf N}^\Lambda ;\, \vert \alpha \vert \in A\}, } if $A\subset{\bf N}$. \par If $j\in \Lambda ^J$, $k\in \Lambda ^K$ as above, then \ekv{3.7} { \prod_{\nu \in J}Z_{j(\nu )}=Z^\alpha ,\ \prod_{\mu \in K}Z^*_{k(\mu )}=(Z^*)^\beta , } where $\alpha =m(j)$, $\beta =m(k)$, and where we use standard multiindex notation, $Z^\alpha =\prod_{\lambda \in\Lambda }Z_\lambda ^{\alpha (\lambda )}$. Conversely for a given $\alpha \in {\bf N}_N^\Lambda $, the number of $j\in \Lambda ^J$ with $m(j)=\alpha $ is equal to ${N!\over\alpha !}={\vert \alpha \vert !\over\alpha !}$. For a typical term in (3.3), write \ekv{3.8} { \prod_{\mu \in K_p}\partial _{x_{k(\mu )}}=\partial _x^{\beta _p},\ \prod_{\nu \in J_p}\partial _{x_{j(\nu )}}=\partial _x^{\alpha _p}, } and similarly with $\partial _x$ replaced by $Z$ or $Z^*$. Then \ekv{3.9} {\alpha =\alpha _0+..+\alpha _{P+1},\ \beta =\beta _0+..+\beta_{P+1},\hbox{ with }\alpha _p\ne 0\ne\beta _p\hbox{ when }1\le p\le P. } Conversely, for such a decomposition of $\alpha =m(j)$, we consider the decomposition $\alpha (\lambda )=\alpha _0(\lambda )+..+\alpha _{P+1}(\lambda )$ for every $\lambda \in\Lambda $, and see that there are $\displaystyle {\alpha !\over \alpha _0!..\alpha _{P+1}!}$ corresponding partitions of $J$ into $J_0\cup ..\cup J_{P+1}$. The equality of the expressions in (3.1) and in (3.3) becomes \eeekv{3.10} { {Z^\alpha \over \alpha !}\circ f\circ {(Z^*)^\beta \over \beta !}= } { \sum_{P\ge 0}{1\over P!}\sum_{{{\alpha =\alpha _0+..+\alpha _{P+1},}\atop{\beta =\beta _0+..+\beta _{P+1},}}\atop{\alpha _j,\beta _j\ne 0{\rm \, for\, }1\le j\le P.}} {(Z^*)^{\alpha _0}\over \alpha _0!}h^{{1\over 2}(\vert \alpha _{P+1}\vert +\vert \beta _{P+1}\vert )}{\partial _x^{\alpha _{P+1}+\beta _{P+1}}f\over \alpha _{P+1}!\beta _{P+1}!}\times }{\hskip 5cm \prod_{p=1}^P (h^{-1+{1\over 2}(\vert \alpha _p\vert +\vert \beta _p\vert )}{\partial _x^{\alpha _p+\beta _p}\phi \over \alpha _p!\beta _p!}){Z^{\beta _0}\over \beta _0!}.} \par We shall transform our expression further by using non-commutative expansions of the tensors appearing in (3.3), (3.10). For this, it seems easier to work with (3.3), and we assume that $f$ is 0 standard, or possibly $M$ standard, depending on $M$ additional indices. For simplicity, we write $$\prod_{\mu \in K_0}Z^*_{k(\mu )}=Z^*_{k\vert K_0},\ \ (\prod_{\mu \in K_p}\partial _{x_{k(\mu )}}\prod_{\nu \in J_p}\partial _{x_{j(\nu )}})\phi =\phi _{k\vert K_p,j\vert J_p}.$$ Now apply Proposition A.1 to one of the tensors: \eekv{3.11} { \phi _{k\vert K_p,j\vert J_p}(x)=\phi _{k\vert K_p,j\vert J_p}(0)+h^{1\over 2}\sum_{\ell\in\Lambda }Z_\ell^*\circ \phi ^{(0)}_{k\vert K_p,j\vert J_p,\ell}(x)+}{\hskip 4cm h^{1\over 2}\sum_{\ell\in\Lambda }\phi ^{(0)}_{k\vert K_p,j\vert J_p,\ell}(x)\circ Z_\ell + h\phi ^{(1)}_{k\vert K_p,j\vert J_p}(x).} When substituting this into (3.3), the effect of the first term of the RHS will be to freeze the corresponding factor to $x=0$. For the contributions of the second term of the RHS of (3.11) in (3.3), we move the $Z_\ell^*$ to the left, until either it joins the factors $Z^*_{k\vert K_0}$ or until it forms a commutator with a $\phi _{k\vert K_q,j\vert J_q}$ or with $f_{k\vert K_{P+1},j\vert J_{P+1}}$, that we denote by $\phi _{k\vert K_q,j\vert J_q,\ell}$ (also for $q=P+1$). In the second case the $\ell$ summation amounts to the contraction of two standard tensors, which produces a standard tensor and an additional power of $h$. For the contribution of the last sum in (3.11), we move the factors $Z_\ell$ as far as possible to the right and repeat the same discussion. The contribution from the last term in (3.11) in (3.3) is simply to introduce an extra power of $h$. The procedure can be iterated a finite number of times, and we see that the general term term in (3.3) becomes a finite sum of terms of the type $$\eqalignno{ \sum_{\ell\in \Lambda ^{L_1\cup ..\cup L_{Q+1}}\atop r\in \Lambda ^{R_1\cup ..\cup R_{Q+1}}}&h^X Z_{k\vert K_0}^*Z^*_{\ell\vert L_1\cup ..\cup L_{Q+1}}\circ \Phi ^{(1)}_{k\vert K_1,j\vert J_1,\ell\vert L_1,r\vert R_1}(x)..&(3.12)\cr&\Phi ^{(Q+1)}_{k\vert K_{Q+1},j\vert J_{Q+1},\ell\vert L_{Q+1},r\vert R_{Q+1}}(x)\circ Z_{j\vert J_0}Z_{r\vert R_1\cup ..\cup R_Q}.} $$ Here $K=K_0\cup ..\cup K_{Q+1}$, $J=J_0\cup ..\cup J_{Q+1}$ are partitions and $K_q\ne\emptyset\ne J_q$ for $1\le q\le Q$. $\Phi ^{(q)}$ are standard, $L_q$, $R_q$ are finite disjoint sets, possibly empty, and $$X={1\over 2}\# (L_1\cup ..\cup L_Q)+{1\over 2}\# (R_1\cup ..\cup R_Q)+N+\sum_1^Q({1\over 2}(\# K_q+\# J_q)-1)+{1\over 2}(\# K_{Q+1}+\# J_{Q+1}),$$ where $N\in {\bf N}$, and we have arranged that $\Phi ^{(Q+1)}$ is the factor which contains the contribution from $f$ under the contraction procedure. \par The point with the further Taylor expansions of the term (0.3) is to arrive at terms with constant factors $\Phi ^{(\nu )}$. More precisely, we can introduce a stopping rule, so that we only get terms of the form (3.12) with \ekv{3.13} {\# (K_0\cup L_1\cup ..\cup L_{Q+1})\le A,} \ekv{3.14} {\# (J_0\cup R_1\cup ..\cup R_{Q+1})\le B,} \ekv{3.15} {N\le N_0,} where $A,B,N_0$ are given integers $\ge 0$ with $A\ge \# K$, $B\ge \#J$, and so that the factors $\Phi ^{(\nu )}$ are constant for all terms for which we have strict inequality in all the three equations (3.13--15). \par We will also need a slight variation of the arguments above. Assume for simplicity that $f=1$. Then from (3.3), we see that (3.1) takes the the form \ekv{3.16} { \sum_{P\ge 0}{1\over P!}\sum_{{J=J_0\cup ..\cup J_P\atop K=K_0\cup ..\cup K_P}\atop{{\rm partitions\, with}\atop J_p\ne\emptyset\ne K_p{\rm \, for\,}1\le p\le P}} Z^*_{k\vert K_0}\prod_{p=1}^P(h^{-1+{1\over 2}(\# J_p+\# K_p)}\partial _{x_{k\vert K_p}}\partial _{x_{j\vert J_p}}\phi (x))Z_{j\vert J_0}. } We now want the coefficients to the left, so we move all the factors $\partial _{x_{k\vert K_p}}\partial _{x_{j\vert J_p}}\phi(x)$ to the left, taking into account the commutator terms. Then the expression (3.1) becomes: \ekv{3.17} {\sum_{P\ge 0}\sum_{{J=J_0\cup ..\cup J_P\atop K=K_0\cup ..\cup K_P}\atop{{\rm partitions\, with}\atop J_p\ne\emptyset\ne K_p{\rm \, for\,}1\le p\le P}}C^{J_0,..,J_P}_{K_0,..,K_P}\prod_{p=1}^P(h^{-1+{1\over 2}(\# J_p+\# K_p)}\partial _{x_{k\vert K_p}}\partial _{x_{j\vert J_p}}\phi (x)) Z^*_{k\vert K_0}Z_{j\vert J_0}. } Here the combinatorial coefficients $C^{\cdots }_{\cdots }$ are independent of $\Lambda $. %\vfill\eject \bigskip \centerline{\bf 4. Study of $\displaystyle ( {1\over \alpha !}(Z^*)^\alpha (e^{-\phi /h})\vert {1\over \beta !}(Z^*)^\beta (e^{-\phi /h})) $.} \medskip \par After adding a $h$-dependent constant to $\phi $, we assume that \ekv{4.1} {\int e^{-\phi (x)/h}dx=1.} We want to study the matrix formed by the scalarproducts in the title of this section, for $\vert \alpha \vert , \vert \beta \vert \le N_0$, for some $N_0\in{\bf N}$. Equivalently, we want to study, \ekv{4.2} { (Z^*_{k\vert K}(e^{-\phi /2h})\vert Z^*_{j\vert J}(e^{-\phi /2h})), } for $J=\{ 1,..,N\}$, $K=\{ 1,..,M\}$, $0\le N,M\le N_0$, $k\in\Lambda ^K$, $j\in\Lambda ^J$. Here we use the notation of section 3. We write this as \ekv{4.3} { (Z_{j\vert J}Z^*_{k\vert K}(e^{-\phi /2h})\vert e^{-\phi /2h}), } and apply ( 3.12), with $f=1$, in which case the factor $\Phi ^{(Q+1)}$ drops out. Since $Z(e^{-\phi /2h})=0$, we can further restrict our attention to the terms with $K_0,L_q,J_0,R_q$ all empty, and it follows that (4.2) is a finite sum of terms of the type \ekv{4.4} { h^X(\Phi ^{(1)}_{k\vert K_1,j\vert J_1}..\Phi ^{(Q)}_{k\vert K_Q,j\vert J_Q}e^{-\phi /2h}\vert e^{-\phi /2h}). } Here $K=K_1\cup ..\cup K_Q$, $J=J_1\cup ..\cup J_Q$ are partitions with $K_q\ne \emptyset\ne J_q$ for all $q$. Further, \ekv{4.5} {X=N+\sum_1^Q ({1\over 2}(\# K_q+\# J_q)-1),\ N\in [0,N_1]\cap {\bf N}, } where $N_1$ is any fixed sufficiently large integer $\ge 0$, and as we saw in section 3, we may arrange that $\Phi ^{(\nu )}$ are constant for the terms with $N0$ such that the following holds for $h>0$ small enough: \smallskip \par\noindent (A) If $-C\le z\le (N+1)\lambda _{\min }(\phi ''(0))-1/C$, then (Gr($N$,0)) has a unique solution $(u,u_-)\in{\cal H}_1\times \ell^2$ for every $(v,v_+)\in{\cal H}_{-1}\times\ell^2$, and \ekv{5.10} { \Vert u\Vert _{{\cal H}_1}+\vert u_-\vert _2\le \widetilde{C}(\Vert v\Vert _{{\cal H}_{-1}}+\vert v_+\vert _2). } \smallskip \par\noindent (B) If $-C\le z\le (N+2)\lambda _{\min }(\phi ''(0))-1/C$, then (Gr($N$,1)) has a unique solution $(u,u_-)\in{\cal H}_1\times \ell^2$ for every $(v,v_+)\in{\cal H}_{-1}\times \ell^2$ and (5.10) holds. \smallskip \par Here $\lambda _{\min}(\phi ''(0))>0$ denotes the smallest eigenvalue of $\phi ''(0)$ and ${\cal H}_{\pm 1}={\cal H}_{\pm 1}^{(\nu )}$ in case $\nu $, $\ell^2=\ell^2({\bf N}^\Lambda _{[0,N]})$ in the case $\nu =0$, $\ell^2=\ell^2(\Lambda )\otimes\ell^2({\bf N}^\Lambda _{[0,N]})$, when $\nu =1$.\rm\medskip \par We shall prove the proposition following the inductive scheme $${\rm Gr\,}(k,1)\to{\rm Gr\,}(k+1,0)\to{\rm Gr\,}(k+1,1),$$ where we start by considering ${\rm Gr}(-1,1)$, which by definition is the problem \ekv{5.11} { (\Delta _\phi ^{(1)}-z)u=v,\ u\in{\cal H}_1,\, v\in{\cal H}_{-1}. } \medskip \par\noindent \bf Lemma 5.2. \it For all $C\ge 1$, $-C\le z\le \lambda _{\min}(\phi ''(0))-1/C$, and $h$ sufficiently small, depending on $C$, the problem (5.11) has a unique solution $u\in{\cal H}_{1}$, for every $v\in{\cal H}_{-1}$. Moreover, \ekv{5.12} {\Vert u\Vert _{{\cal H}_1}\le\widetilde{C}\Vert v\Vert _{{\cal H}_{-1}},} where $\widetilde{C}>0$ depends on $C$ but not on $z,h,\Lambda $.\rm\medskip \par\noindent \bf Proof. \rm Recall that $\Delta _\phi ^{(1)}=1\otimes\Delta _\phi ^{(0)}+\phi ''(x)$ and consider first the simplified problem, \ekv{5.13} {(1\otimes\Delta _\phi ^{(0)}+\phi ''(0)-z)u=v,\ u\in{\cal H}_1,\, v\in{\cal H}_{-1}.} If $u$ solves (5.13), take the scalar product of this equation with $u$ and get $$\eqalign{& \Vert v\Vert _{{\cal H}_{-1}}\Vert u\Vert _{{\cal H}_1}\ge (v\vert u)=((1\otimes\Delta _\phi ^{(0)}+\phi ''(0)-z)u\vert u)\cr&\ge \epsilon ((1\otimes \Delta _\phi ^{(0)}+1)u\vert u)+((\phi ''(0)-z-\epsilon )u\vert u)\ge \epsilon \Vert u\Vert _{{\cal H}_1}^2 ,}$$ for $\epsilon >0$ small enough, so \ekv{5.14} { \Vert u\Vert _{{\cal H}_1}\le C\Vert v\Vert _{{\cal H}_{-1}}. } This gives injectivity and the analogue of (5.12) for the problem (5.13). Since $(1\otimes\Delta _\phi ^{(0)}+\phi ''(0)-z)$ is a bounded selfadjoint operator ${\cal H}_1\to{\cal H}_{-1}$, it is also surjective, so (5.13) is uniquely solvable and satisfies (5.14). To get the lemma it suffices to use that $\phi ''(x)-\phi ''(0)={\cal O}(h^{1/2}):{\cal H}_{1}\to{\cal H}_{-1}$.\hfill{$\#$} \medskip \par The preceding lemma gives well-posedness for ${\rm Gr\,}(-1,1)$ in the appropriate range. Let us now perform the step ${\rm Gr\,}(-1,1)\to{\rm Gr\,}(0,0)$, so consider \ekv{5.15} { \cases{(\Delta _\phi ^{(0)}-z)u+R_-^{0,0}u_-=v\cr R_+^{0,0}u=v_+,} } with $v\in{\cal H}_{-1}$, $u\in{\cal H}_1$, $u_-,v_+\in{\bf C}$, $R_+^{0,0}u=(u\vert e^{-\phi /2h})$. We let $z$ be in the range of the lemma above, and we first prove uniqueness in (5.15). Let $v=0$, $v_+=0$ in (5.15). Since $d_\phi R_-^{0,0}=0$, $d_\phi \Delta _\phi ^{(0)}=\Delta _\phi ^{(1)}d_\phi $, we get by applying $d_\phi $ to the first equation in (5.15): \ekv{5.16} {(\Delta _\phi ^{(1)}-z)d_\phi u=0.} Here we only know apriori that $d_\phi u\in L^2$, so we cannot apply Lemma 5.2 directly. However, it is easy and standard to show that every $L^2$ solution $w$ of $(\Delta ^{(1)}_\phi -z)w=0$, has to belong to ${\cal S}$ and in particular to ${\cal H}_1$. Consequently, we can apply Lemma 5.2 and conclude that $d_\phi u=0$. Since $d_\phi =h^{1/2}e^{-\phi /2h}\circ d\circ e^{\phi /2h}$, it follows that $u=\lambda e^{-\phi /2h}$ for some constant $\lambda \in{\bf R}$. Using also that $v_+=0$ in (5.15), we see that $\lambda =0$, so $u=0$ and then $u_-=0$, and we have proved uniqueness for solutions of (5.15). Define $z_0\in [0,\infty [$ by \ekv{5.17} { z_0=\inf_{u\in{\cal H}_1\cap (e^{-\phi /2h})^\perp\atop \Vert u\Vert =1}(\Delta _\phi ^{(0)}u\vert u). } Since the inclusion map ${\cal H}_1\to L^2$ is compact, there exists $u_0\in{\cal H}_1\cap(e^{-\phi /2h})^\perp$ with $\Vert u_0\Vert =1$, such that $z_0=(\Delta _\phi ^{(0)}u_0\vert u_0)$, i.e. $((\Delta _\phi ^{(0)}-z_0)u_0\vert u_0)=0$, while $((\Delta _\phi ^{0)}-z_0)u\vert u)\ge 0$ for general $u\in{\cal H}_1\cap (e^{-\phi /2h})^\perp$. It follows that $(\Delta _\phi ^{(0)}-z_0)u_0=\mu e^{-\phi /2h}$ for some $\mu \in{\bf R}$ and since $u_0\perp e^{-\phi /2h}$ and $e^{-\phi /2h}\in{\rm Ker\,}\Delta _\phi ^{(0)}$, we see that $\mu =0$. Hence $(\Delta _\phi ^{(0)}-z_0)u_0=0$, so $u=u_0$, $u_-=0$ is a solution of (5.15) with $v=0$, $v_+=0$ and $z=z_0$. Since we know that (5.15) is injective for $0\le z\le \lambda _{\rm \min}(\phi ''(0))-1/2C$, for $h$ small enough depending on $C$, we conclude that \ekv{5.18} { z_0\ge \lambda _{\rm min}(\phi ''(0))-{1\over 2C}. } \par Let us now restrict the attention to $-C\le z\le \lambda _{\rm min}(\phi ''(0))-1/C$ and derive an apriori estimate for solutions to (5.15). Let first $v_+=0$ in (5.15), and take the scalar product of the first equation there with $u$, and use that $(R_-^{0,0}u_-\vert u)=(u_-\vert R_+^{0,0}u)=0$. We get \ekv{5.19} { ((\Delta _\phi ^{(0)}-z)u\vert u)=(v\vert u). } With $\delta >0$ small enough, write $$\Delta _\phi ^{(0)}-z=\delta \Delta _\phi ^{(0)}+(1-\delta )(\Delta _\phi ^{(0)}-z_0)+(1-\delta )(z_0-{z\over 1-\delta }),$$ and get $$((\Delta _\phi ^{(0)}-z)u\vert u)\ge \delta (\Delta _\phi ^{(0)}u\vert u)+(1-\delta )(z_0-{z\over 1-\delta })\Vert u\Vert ^2\ge \delta \Vert u\Vert _{{\cal H}_1}^2.$$ Hence from (5.19), we get $$\Vert u\Vert _{{\cal H}_1}^2\le \widetilde{C}\Vert v\Vert _{{\cal H}_{-1}}\Vert u\Vert _{{\cal H}_1},$$ \ekv{5.20} { \Vert u\Vert _{{\cal H}_1}\le \widetilde{C}\Vert v\Vert _{{\cal H}_{-1}}, } for solutions of (5.15) with $v_+=0$. \par Now take the scalar product of the first equation in (5.15) with $R_-^{0,0}u_-$ and get $$-z(u\vert R_-^{0,0}u_-)+\vert u_-\vert ^2=\overline{u}_-(v\vert e^{-\phi /h}).$$ With (5.20), this gives $\vert u_-\vert ^2\le\widehat{C}\Vert v\Vert _{{\cal H}_{-1}}\vert u_-\vert $ and hence \ekv{5.21} {\vert u_-\vert \le \widetilde{C}\Vert v\Vert _{{\cal H}_{-1}},} where we let $\widetilde{C}$ denote a new constant in every new formula. \par If $v_+\ne 0$, consider $\widetilde{u}:=u-v_+e^{-\phi /2h}$, which solves \ekv{5.22} { \cases{(\Delta _\phi ^{(0)}-z)\widetilde{u}+R_-^{0,0}u_-= v+zv_+e^{-\phi /2h}\cr R_+^{0,0}\widetilde{u}=0. } } Applying (5.20), (5.21) to this system, we get $$\Vert \widetilde{u}\Vert _{{\cal H}_1}+\vert u_-\vert \le\widetilde{C}(\Vert v\Vert _{{\cal H}_{-1}}+\vert v_+\vert ),$$ leading to \ekv{5.23} { \Vert u\Vert _{{\cal H}_1}+\vert u_-\vert \le\widetilde{C}(\Vert v\Vert _{{\cal H}_{-1}}+\vert v_+\vert ) ,} for solutions of (5.15). Since this problem is selfadjoint, we also have existence and we have proved the proposition for (Gr(0,0)). \par Les us now prove that if for some $N\in{\bf N}$ the proposition is valid for (Gr($N$,0)) then it is valid for (Gr($N$,1)). So we assume for a fixed $N$ that (A) holds for all $C$ with $h>0$ small enough depending on $C$, and we want to prove (B) with the same $N$. Using again that $\phi ''(x)-\phi ''(0)={\cal O}(h^{1/2}):{\cal H}_1\to{\cal H}_{-1}$, we see that it suffices to treat the simplified problem \ekv{5.24} { \cases{(1\otimes\Delta _\phi ^{(0)}+\phi ''(0)-z)u+R_-^{N,1}u_-=v\cr R_+^{N,1}u=v_+.} } Since we have (A), for the chosen value of $N$, we know from the preceding discussion that \ekv{5.25} { \inf_{w\in{\cal H}_1,\, \Vert w\Vert =1\atop R_+^{N,0}w=0}(\Delta _\phi ^{(0)}w\vert w)\ge (N+1)\lambda _{\min}(\phi ''(0))-{1\over 2C}, } for $h>0$ small enough depending on $C$. Consider (5.24) in the case $v_+=0$. Then $R_+^{N,0}u_j=0$ for each component $u_j$ of $u$ and consequently $$((1\otimes\Delta _\phi ^{(0)})u\vert u)\ge ((N+1)\lambda _{{\rm min}}-{1\over 2C})\Vert u\Vert ^2.$$ Since $(\phi '(0)u\vert u)\ge \lambda _{\rm min}\Vert u\Vert ^2$, we get \ekv{5.26} { ((1\otimes\Delta _\phi ^{(0)}+\phi ''(0)-z)u\vert u)\ge {1\over 2C}\Vert u\Vert ^2, } for $z$ in the range of values of (B). As before, this leads to \ekv{5.27} { ((1\otimes\Delta _\phi ^{(0)}+\phi ''(0)-z)u\vert u)\ge \delta \Vert u\Vert _{{\cal H}_1}^2, } for some $\delta >0$. Take the scalar product of the first equation in (5.24) with $u$, and use that $(R_-^{N,1}u_-\vert u)=(u_-\vert R_+^{N,1}u)=0$. Then $$\delta \Vert u\Vert _{{\cal H}_1}^2\le \Vert v\Vert _{{\cal H}_{-1}}\Vert u\Vert _{{\cal H}_1},$$ which gives, \ekv{5.28} {\Vert u\Vert _{{\cal H}_1}\le\widetilde{C}\Vert v\Vert _{{\cal H}_{-1}}} for solutions of (5.24) with $v_+=0$, when $z$ is in the range of (B). \par We next want to take the scalarproduct with $R_-^{N,1}u_-$, and as a preparation we need to establish two results about $R_\pm$. \medskip \par\noindent \bf Lemma 5.3. \it $R_+^{N,0}R_-^{N,0}$ is ${\cal O}(1):\ell^2({\bf N}^\Lambda _{[0,N]})\to\ell^2({\bf N}^\Lambda _{[0,N]})$ and has a uniformly bounded inverse. Moreover, if $r_P:\ell^2({\bf N}^\Lambda _{[0,N]})\to\ell^2({\bf N}^\Lambda _P)$ is the natural restriction operator, then for $0\le P,Q\le N$. \ekv{5.29} { r_PR_+^{N,0}R_-^{N,0}r_Q^*\sim\sum_{\nu =0}^\infty h^{\nu +{1\over 2}\vert P-Q\vert }M_{\nu ;P,Q}, } in ${\cal L}(\ell^2({\bf N}^\Lambda _Q),\ell^2({\bf N}_P^\Lambda ))$ uniformly with repect to $\Lambda $. Here \ekv{5.30} { M_{0;P,P}^{(N)}=\phi ''(0)\odot ..\odot\phi ''(0). } \rm\medskip \par\noindent \bf Proof. \rm For simplicity, we work with the equivalent operators $\widetilde{R}_{\pm}^{N,0}$ between $L^2({\bf R}^\Lambda )$ and $\ell^2_b(\Lambda ^0\cup\Lambda ^1\cup ..\cup\Lambda ^N)$, where the subscript $b$ indicates that we take the "Bosonic" subspace of permutation invariant elements of $\ell^2$. Then the matrix of $r_P\widetilde{R}_+^{N,0} \widetilde{R}_-^{N,0}r_Q^*$ is given by \ekv{5.31} {{1\over\sqrt{P!Q!}}(Z^*_{p\vert {\cal P}}(e^{-\phi /2h})\vert Z^*_{q\vert {\cal Q}}(e^{-\phi /2h})),} with ${\cal P}=\{ 1,..,P\}$, ${\cal Q}=\{ 1,..,Q\}$, $p\in\Lambda ^{\cal Q}$, $p\in\Lambda ^{\cal P}$. The uniform asymptotic expansion (5.29) then follows from Proposition 4.1. Moreover, the matrix $\widetilde{M}_{0,P,P}^{(N)}$ (corresponding to $M^{(N)}_{0,P,P}$) has the elements \ekv{5.32} { {1\over P!}\sum_{\pi \in{\rm Perm\,}(K)}\prod_{\nu \in{\cal P}}\phi _{p(\nu ),q(\pi (\nu ))}''(0),} which has the same action on $\ell^2_b(\Lambda ^P)$ as the matrix \ekv{5.33} { \prod_{\nu \in{\cal P}}\phi _{p(\nu ),q(\nu )}''(0), } which is simply the matrix $\phi ''(0)\otimes ..\otimes \phi ''(0)$.\hfill{$\#$} \medskip \par By tensoring all the spaces with $\ell^2(\Lambda )$, we get the obvious analogue of Lemma 5.3 for $R_+^{N,1}R_-^{N,1}$. It also follows from Lemma 5.3, that $R_-^{N,0}$ is uniformly ${\cal O}(1):\ell^2\to L^2$. Consequently $R_+^{N,0}={\cal O}(1):L^2\to\ell^2$, and we have the corresponding facts for $R_\pm^{N,1}$. This can be strengthened: \medskip \par\noindent \bf Lemma 5.4. \it $R_-^{N,0}$ is uniformly bounded: $\ell^2({\bf N}^\Lambda _{[0,N]})\to{\cal H}_1$. Consequently $R_+^{N,0}$ is uniformly bounded ${\cal H}_{-1}\to\ell^2({\bf N}^\Lambda _{[0,N]})$. \medskip \par\noindent \bf Proof. \rm Again we think it is more convenient to work with the equivalent operator $\widetilde{R}_-^{N,0}$. Let $1\le M\le N$, $u\in\ell_b^2(\Lambda ^M)$ and consider \ekv{5.34} { Z_j\widetilde{R}_-^{N,0}u=\sum_{m\in\Lambda ^{{\cal M}}}{1\over\sqrt{M!}}Z_jZ^*_{m\vert {\cal M}}(e^{-\phi /2h})u(m), } where ${\cal M}=\{ 1,..,M\}$. We apply the expression (3.17) to obtain \ekv{5.35} { Z_j\widetilde{R}_-^{N,0}u=\sum_{m\in\Lambda ^{{\cal M}}}\sum_{{{{\cal M}=M_0\cup M_1}\atop{{\rm partition\, with}}}\atop {M_1\ne \emptyset}}C_{M_0,M_1}h^{-{1\over 2}+{1\over 2}\#M_1}\partial _{x_j}\partial _{x_{m\vert M_1}}\phi (x)Z^*_{m\vert M_0}(e^{-\phi /2h})u(m), } where $C_{M_0,M_1}$ is independent of $\Lambda $, and equal to $1/\sqrt{M!}$ when $\#M_1=1$. For $u\in \ell_b^2(\Lambda ^M)$, $v\in \ell_b^2(\Lambda ^P)$, $1\le P,M\le N$, we get \ekv{5.36} { \sum_j(Z_j\widetilde{R}_-^{N,0}u\vert Z_j\widetilde{R}_-^{N,0}v)=(B^{P,M}u\vert v), } where $B^{P,M}$ is given by a matrix $B_{p,m}^{P,M}$, $p\in\Lambda ^P$, $m\in\Lambda ^M$, which is a finite linear combination of terms \ekv{5.37} { h^{-1+{1\over 2}\#M_1+{1\over 2}\# P_1}(Z_{p\vert P_0}\sum_j(\partial _{x_{p\vert P_1}}\partial _{x_j} \phi )(\partial _{x_j}\partial _{x_{m\vert M_1}}\phi )Z^*_{m\vert M_0}(e^{-\phi /2h})\vert e^{-\phi /2h}),} where ${\cal M}=M_0\cup M_1$, ${\cal P}=P_0\cup P_1$ are partitions with $M_1\ne\emptyset\ne P_1$. Here $$\Phi _{p\vert P_1,m\vert M_1}:=\sum_j(\partial _{x_{p\vert P_1}}\partial _{x_j} \phi )(\partial _{x_j}\partial _{x_{m\vert M_1}}\phi )$$ is a standard tensor, being the contraction of two standard tensors of size $1+\#P_1$ and $1+\#M_1$, with at least one of the sizes $\ge 2$ (cf. Lemma 8.2). \par As in sections 3,4, in particular the discussion leading to (3.12-15), we see that (5.37) is a finite sum of terms \ekv{5.38} { h^X(\Phi ^{(1)}_{p\vert \widetilde{P}_1,m\vert \widetilde{M}_1}..\Phi _{p\vert \widetilde{P}_Q,m\vert \widetilde{M}_Q }^{(Q)}e^{-\phi /2h}\vert e^{-\phi /2h}),} where ${\cal P}=\widetilde{P}_1\cup ..\cup\widetilde{P}_Q$, ${\cal M}=\widetilde{M}_1\cup ..\cup\widetilde{M}_Q$ are partitions with $\widetilde{P}_q,\widetilde{M}_q\ne \emptyset$, $\widetilde{P}_1\supset P_1$, $\widetilde{M}_1\supset M_1$. $\Phi ^{(q)}$ are standard tensors and $$X=\widetilde{N}+\sum_1^Q({1\over 2}(\#\widetilde{M}_q+\#\widetilde{P}_q)-1),\ \widetilde{N}\in[0,N_1]\cap{\bf N}. $$ Here we can fix any $N_1\in{\bf N}$, and the $\Phi ^{(q)}$ are independent of $x$, when $\widetilde{N}0$ as large as we like). \par As before we conclude that $$\inf_{u\in{\cal H}_1,\,\Vert u\Vert =1\atop R_+^{N+1,0}u=0}(\Delta _\phi ^{(0)}u\vert u)\ge (N+2)\lambda _{\min}(\phi ''(0))-{1\over 2C},$$ for every $C>0$ when $h>0$ is small enough depending on $C$. By repeating earlier arguments, we obtain the apriori estimate (5.10) for solutions to Gr($N$+1,0), as well as existence of such solutions for arbitrary $v\in{\cal H}_{-1}$ and $v_+\in\ell^2$. In other words, we get part (A) of the proposition with $N$ replaced by $N+1$ and this completes the inductive proof of Proposition 5.1. \medskip\it Remark 5.5. \rm Let us compute $(\Delta _\phi ^{(0)}\widetilde{R}_-^{N,0}u\vert \widetilde{R}_-^{N,0}v)$ to leading order for $u,v\in\ell_b^2(\Lambda ^0\cup ..\cup \Lambda ^N)$, i.e. modulo ${\cal O}(1)h^{1/2}\vert u\vert _2\vert v\vert _2$. The proof of Lemma 5.4 shows that the searched expression involves a block diagonal matrix, so we may assume that $u,v\in\ell_b^2(\Lambda ^{{\cal P}})$, ${\cal P}=\{ 1,..,P\}$, for $1\le P\le N$. (The case $P=0$ will give 0.) Then if $\equiv$ indicates equality modulo ${\cal O}(1)h^{1/2}\vert u\vert _2\vert v\vert _2$, we get $$\eqalign{ &(\Delta _\phi ^{(0)}\widetilde{R}_-^{N,0}u\vert \widetilde{R}_-^{N,0}v)=\sum_{j\in\Lambda }(Z_j\widetilde{R}_-^{N,0}u\vert Z_j \widetilde{R}_-^{N,0}v)\equiv\cr & {1\over P!}\sum_{j\in\Lambda }\sum_{\widehat{p},\widehat{q}= 1}^P\sum_{p,q\in\Lambda ^{{\cal P}}}\phi ''_{q(\widehat{q}),j}(0)\phi ''_{j,p(\widehat{p})}(0)(Z^*_{p\vert {\cal P}\setminus\{\widehat{p}\}}(e^{-\phi /2h}) \vert Z^*_{q\vert {\cal P}\setminus\{ \widehat{q}\}}(e^{-\phi /2h}))u(p)\overline{v(q)}. }$$ Using that $u,v\in\ell_b^2$, we can reduce the sum to the case $\widehat{p}=\widehat{q}=1$, and get $$ \eqalign{ &{P^2\over P!}\sum_{j\in\Lambda } \sum_{p,q\in\Lambda ^{{\cal P}}}\phi ''_{q(1),j}(0)\phi ''_{j,p(1)}(0)(Z^*_{p\vert {\cal P}\setminus\{ 1\} }(e^{-\phi /2h})\vert Z^*_{q\vert {\cal P}\setminus\{ 1\}}(e^{-\phi /2h}))u(p)\overline{v(q)}\cr & ={P^2\over P!}\sum_{p,q\in\Lambda ^{{\cal P}}}(\phi ''(0)^2)_{q(1),p(1)}(Z^*_{p\vert {\cal P}\setminus\{ 1\}}(e^{-\phi /2h})\vert Z^*_{q\vert {\cal P}\setminus\{ 1\}}(e^{-\phi /2h}))u(p)\overline{v(q)}\cr &\equiv {P^2\over P!}\sum_{p,q\in\Lambda ^{{\cal P}}}(\phi ''(0)^2)_{q(1),p(1)}\sum_{\pi \in{\rm Perm}(\{ 2,..,P\} )}(\prod_{\nu =2}^P \phi ''_{p(\nu ),q(\pi (\nu ))}(0))u(p)\overline{v(q)}= \cr & P\sum_{p,q\in\Lambda ^{{\cal P}}}(\phi ''(0)^2)_{q(1),p(1)}\prod_{\nu =2}^P (\phi ''_{p(\nu ), q(\nu )}(0))u(p)\overline{v(q)}=(P\phi ''(0)^2\otimes \phi ''(0)\otimes ..\otimes \phi ''(0)u\vert v)_{\ell^2} \cr& \hskip 2cm =(P(\phi ''(0)\otimes 1\otimes ..\otimes 1)(\phi ''(0)^{1\over 2}\otimes ..\otimes \phi ''(0)^{1\over 2})u\vert (\phi ''(0)^{1\over 2}\otimes ..\otimes \phi ''(0)^{1\over 2})v), }$$ where we again used that $u,v\in\ell_b^2$. Using this property once more, we can replace $P(\phi ''(0)\otimes 1\otimes ..\otimes 1)$ by the more suggestive expression \ekv{5.53}{\Phi _P:=\phi ''(0)\otimes 1\otimes ..\otimes 1+1\otimes \phi ''(0)\otimes 1 ..\otimes 1+..+1\otimes ..\otimes 1\otimes \phi ''(0).} If $\lambda _1,..,\lambda _{\#\Lambda }$ denote the eigenvalues of $\phi ''(0)$, then the eigenvalues of (5.53) are of the form \ekv{5.54} {\sum_{\nu =1}^P\lambda _{p(\nu )},\ p\in\{ 1,..,\#\Lambda \}^{\cal P}.} \par Summing up the discussion, for $u\in \ell_b^2(\Lambda ^{{\cal P}})$, $v\in\ell_b^{{\cal Q}}$, ${\cal P}=\{ 1,..,P\}$, ${\cal Q}=\{ 1,..,Q\}$, we have \eekv{5.55} {(\Delta _\phi ^{(0)}\widetilde{R}_-^{N,0}u\vert \widetilde{R}_-^{N,0}v)}{={\cal O}(h^{1\over 2})\vert u\vert _2\vert v\vert _2+\cases{ 0,\hbox{ if }P\ne Q\cr (\Phi _P(\phi ''(0)^{1\over 2}\otimes ..\otimes \phi ''(0)^{1\over 2})u\vert (\phi ''(0)^{1\over 2}\otimes ..\otimes \phi ''(0)^{1\over 2})v),\ P=Q. }} This should be compared with the following consequence of section 4: \eekv{5.56} {(\widetilde{R}_-^{N,0}u\vert \widetilde{R}_-^{N,0}v)}{={\cal O}(h^{1\over 2})\vert u\vert _2\vert v\vert _2+\cases{ 0,\hbox{ if }P\ne Q\cr ((\phi ''(0)^{1\over 2}\otimes ..\otimes \phi ''(0)^{1\over 2})u\vert (\phi ''(0)^{1\over 2}\otimes ..\otimes \phi ''(0)^{1\over 2})v),\ P=Q. }} %\vfill\eject \bigskip \centerline{\bf 6. Asymptotics of the solutions of the Grushin problems.} \medskip \par We first work with the scalar case and denote by ${\cal L}_j$ the span of all ${(Z^*)^\alpha \over\alpha !}(e^{-\phi /2h})$, $\vert \alpha \vert =j$. Equivalently, ${\cal L}_j$ is equal to $R_-^{N,0}(\ell^2({\bf N}_j^\Lambda ))$, if $j\le N$. If $A$ is a finite subset of ${\bf N}$, we write ${\cal L}_A=\oplus_{j\in A}{\cal L}_j\subset L^2.$ Notice that the orthogonal projection onto ${\cal L}_{[0,N]}$ is given by \ekv{6.1} { R_-^{N,0}(R_+^{N,0}R_-^{N,0})^{-1}R_+^{N,0}. } By section 4 we know that \ekv{6.2} { \Vert R_-^{N,0}v_+\Vert \sim\vert v_+\vert ,\ u\in\ell^2({\bf N}_{[0,N]}^\Lambda ). } We can identify ${\cal L}_j$ with $\ell^2({\bf N}_j^\Lambda )$ by means of $r_jR_+^{N,0}$, where $r_j:\ell^2({\bf N}_{[0,N]}^\Lambda )\to\ell^2({\bf N}_j^\Lambda )$ is the natural restriction map, and again by section 4 we know that \ekv{6.3} { \vert R_+^{N,0}u\vert _2\sim \Vert u\Vert ,\ u\in{\cal L}_{[0,N]}. } \par We have the decomposition \ekv{6.4} { L^2({\bf R}^\Lambda )={\cal L}_0\oplus ..\oplus{\cal L}_N\oplus {\cal L}_{[0,N]}^\perp,\ u=u_0+..+u_N+u_{N+1}\in L^2, } and correspondingly \ekv{6.5} {\Vert u\Vert ^2\sim\sum_0^{N+1}\Vert u_j\Vert^2. } For $j\le N$, the projection onto ${\cal L}_j$ is given by $$\Pi _j=R_-^{N,0}r_j^*r_j{(R_+^{N,0}R_-^{N,0})}^ {-1} R_+^{N,0}.$$ Lemma 5.4 and (6.2) imply that \ekv{6.6} { \Vert u\Vert _{{\cal H}_1}\le{\cal O}(1)\Vert u\Vert ,\ u\in{\cal L}_{[0,N]}, } and the same lemma with (6.3) implies that \ekv{6.7} { \Vert u\Vert \le{\cal O}(1)\Vert u\Vert _{{\cal H}_{-1}},\ u\in{\cal L}_{[0,N]}. } In other words, the norms of ${\cal H}_1$, ${\cal H}_{-1}$ and $L^2$ are (uniformly) equivalent on ${\cal L}_{[0,N]}$, and we also know that the projections (6.1) and $\Pi _j$ are bounded in these spaces. \par We are interested in the block matrix of $\Delta _\phi ^{(0)}$, viewed as an operator $$\eqalignno{ \Delta _\phi ^{(0)}:&{\cal L}_0\oplus{\cal L}_1\oplus ..\oplus{\cal L}_N\oplus ({\cal H}_1\cap{\cal L}_{[0,N]}^\perp)\to&(6.8) \cr&{\cal L}_0\oplus{\cal L}_1\oplus ..\oplus{\cal L}_N\oplus ({\cal H}_{-1}\cap{\cal L}_{[0,N]}^\perp) .}$$ (3.12) shows that for $j\in\Lambda $, ${\cal M}=\{ 1,..,M\}$, $0\le M\le N$, $m\in\Lambda ^{{\cal M}}$: \eekv{6.9} {Z_jZ_{m\vert {\cal M}}^*(e^{-\phi /2h})=\hbox{ a finite sum of terms of the type}} { \sum_{\ell\in\Lambda ^L}h^XZ_{m\vert M_0}^*Z_{\ell\vert L}^*\circ \Phi _{j,\ell\vert L, m\vert M_1}(x)(e^{-\phi /2h}),\ \# M_0+\# L\le N, } where ${\cal M}=M_0\cup M_1$ is a partition with $M_1\ne\emptyset$, $L$ is finite, and $\Phi $ is standard. Moreover, \ekv{6.10} { X={1\over 2}\#L+\widetilde{N}+{1\over 2}(\# M_1-1),\ 0\le\widetilde{N}\le N_1\in{\bf N}. } Here $N_1$ is any sufficiently large integer and $\Phi _{j,\ell\vert L,m\vert M_1}$ is independent of $x$, when $\# M_0+\# L0$ be constants with $d_{N+1}=1$, such that \ekv{6.27} { d_{j+1}/d_j\in [h^{{1\over 2}},h^{-{1\over 2}}],\ 0\le j\le N, } or satifying the sharper assumption \ekv{6.28} {d_{j+1}/d_j\in [\delta ,1/\delta ], \ 0\le j\le N,} for some $h^{1\over 2}\le \delta \le 1$. Let $\widetilde{d}:\ell^2({\bf N}^\Lambda _{[0,N]})\to\ell^2({\bf N}^\Lambda _{[0,N]})$ be given by the block diagonal matrix ${\rm diag\,}(d_j)_{0\le j\le N}$, with respect to the orthogonal decomposition $\ell^2({\bf N}^\Lambda _{[0,N]})=\oplus_{j=1}^N \ell^2({\bf N}_j^\Lambda )$. Put $$K_+=(R_+R_-)^{-{1\over 2}}R_+,\ K_-=R_-(R_+R_-)^{-{1\over 2}}$$ so that $$K_+^*=K_-,\ K_+K_-=1,\ K_-K_+=\Pi =\Pi _{[0,N]}.$$ Put $d=K_-\widetilde{d}K_++(1-\Pi )$, where we notice that the first term commutes with $\Pi $; $\Pi K_-\widetilde{d}K_+=K_-\widetilde{d}K_+\Pi =K_-\widetilde{d}K_+$. We observe that $d$ and $\widetilde{d}$ are selfadjoint and that $d^{-1}$ corresponds to $\widetilde{d}^{-1}$: $d^{-1}=K_-\widetilde{d}^{-1}K_+ +(1-\Pi )$. \par Consider $\widetilde{d}^{-1}R_+d=\widetilde{d}^{-1} (R_+R_-)^{1\over 2}\widetilde{d}(R_+R_-)^{-{1\over 2}}R_+$. Here we know from Proposition 4.1 that the block matrix elements $((R_+R_-)^{1\over 2})_{j,k}$ are ${\cal O}(h^{\vert j-k\vert /2})$ and it follows that $$\widetilde{d}^{-1}(R_+R_-)^{1\over 2}\widetilde{d}={\cal O}(1):\ell^2\to\ell^2$$ under the assumption (6.27) and that $$\widetilde{d}^{-1}(R_+R_-)^{1\over 2}\widetilde{d}-(R_+R_-)^{1\over 2}={\cal O}(1) {h^{1\over 2}\over \delta }:\ell^2\to\ell^2$$ under the assumption (6.28). We conclude that under the latter assumption \eekv{6.29} {\widetilde{d}^{-1}R_+d-R_+={\cal O}(1){h^{1\over 2}\over \delta }:{\cal H}_{-1}\to\ell^2} {d^{-1}R_- \widetilde{d}-R_-={\cal O}(1){h^{1\over 2}\over \delta }:\ell^2\to{\cal H}_{1}.} Here the second relation follows from the first by duality and in both relations, we are allowed to replace $(\widetilde{d},d)$ be $(\widetilde{d}^{-1}, d^{-1})$. \par Now recall that the ${\cal H}_{\pm 1}$ norms and the $L^2$ norm are all equivalent on ${\cal L}_{[0,N]}$, and consider $$\eqalign{&d^{-1}\Delta _\phi ^{(0)}d-\Delta _\phi ^{(0)}=\cr &K_-(\widetilde{d}^{-1}(R_+R_-)^{-{1\over 2}}R_+\Delta _\phi ^{(0)}R_-(R_+R_-)^{-{1\over 2}}\widetilde{d}-(R_+R_-)^{-{1\over 2}}R_+\Delta _\phi ^{(0)}R_-(R_+R_-)^{-{1\over 2}})K_+\cr &+K_-(\widetilde{d}^{-1}-1)(R_+R_-)^{-{1 \over 2}}R_+\Delta _\phi ^{(0)}(1-\Pi )+(1-\Pi )\Delta _\phi ^{(0)}R_-(R_+R_-)^{-{1\over 2}}(\widetilde{d}-1)K_+.}$$ Here the block matrix element of $(R_+R_-)^{-{1\over 2}}R_+\Delta _\phi ^{(0)}R_-(R_+R_-)^{-{1\over 2}}$ at $(j,k)$ is ${\cal O}(h^{{1\over 2}\vert j-k\vert })$, so $$\eqalign{\widetilde{d}^{-1}(R_+R_-)^{-{1\over 2}}R_+\Delta _\phi ^{(0)}R_-(R_+R_-)^{-{1\over 2}}\widetilde{d}-(R_+R_-)^{-{1\over 2}}R_+\Delta _\phi^{(0)}R_-(R_+R_-)^{-{1\over 2}}\cr={\cal O}(1){h^{1\over 2}\over \delta }:\ell^2\to \ell^2.} $$ Similarly $$\eqalign{(\widetilde{d}^{-1}-1) (R_+R_-)^{-{1\over 2}}R_+\Delta _\phi ^{(0)}(1-\Pi )&={\cal O}(1){h^{{1\over 2}}\over \delta }:L^2\to \ell^2\cr (1-\Pi )\Delta _\phi ^{(0)}R_-(R_+R_-)^{-{1\over 2}}(\widetilde{d}-1)&={\cal O}(1){h^{{1\over 2}}\over \delta }:\ell^2\to L^2, }$$ and we conclude that \ekv{6.30} { d^{-1}\Delta _\phi ^{(0)}d-\Delta _\phi ^{(0)}={\cal O}(1){h^{1\over 2}\over \delta }:{\cal H}_1\to{\cal H}_{-1}. } Define \ekv{6.31} {D=\pmatrix{d &0\cr 0 &\widetilde{d}}={\cal H}_{\pm 1}\times \ell^2({\bf N}^\Lambda _{[0,N]})\to {\cal H}_{\pm 1}\times \ell^2({\bf N}^\Lambda _{[0,N]}).} If \ekv{6.32} { {\cal P}^{N,0}=\pmatrix{ \Delta _\phi ^{(0)}-z&R_-^{N,0}\cr R_+^{N,0}&0}, } then under the assumption (6.27) \ekv{6.33} { D^{-1}{\cal P}^{N,0}D={\cal O}(1):{\cal H}_1\times \ell^2\to{\cal H}_{-1}\times \ell^2, } and if (6.28) holds, then \ekv{6.34} { D^{-1}{\cal P}^{N,0}D-{\cal P}^{N,0}={\cal O}(1)h^{1\over 2}/\delta :{\cal H}_1\times \ell^2\to{\cal H}_{-1}\times \ell^2. } Under the assumptions of Proposition 5.1(A), we introduce \ekv{6.35} { {\cal E}^{N,0}=({\cal P}^{N,0})^{-1}:{\cal H}_{-1}\times \ell^2\to{\cal H}_1\times \ell^2. } Under the assumption (6.27), we have \ekv{6.36} { D^{-1}{\cal E}^{N,0}D={\cal O}(1):{\cal H}_{-1}\times \ell^2\to{\cal H}_{1}\times \ell^2, } (noticing that we have (6.28) with $\delta =Ch^{1/2}$ and $C$ large enough) and if we assume (6.30), then \ekv{6.37} { D^{-1}{\cal E} ^{N,0}D -{\cal E}^{N,0}={\cal O}(1)h^{1\over 2}/\delta . } \par Write \ekv{6.38} { {\cal E}^{N,0}=\pmatrix{E^{N,0}&E_+^{N,0}\cr E_-^{N,0}&E_{-+}^{N,0}}. } We shall derive approximations of $E_{\pm}$, $E_{-+}$, where we sometimes drop the superscript $N,0$ and for that we look for an approximate solution of the system \ekv{6.39} {\cases{(\Delta_\phi^{(0)} -z)u+R_-u_-=0\cr R_+u=v_+.}} Try \ekv{6.40} { u_0=R_-(R_+R_-)^{-1}v_+=:E_+^0v_+, } so that $R_+u_0=v_+$. We will choose $u_-=u_-^0$ in order to satisfy the ${\cal L}_{[0,N]}$ component of the first equation of (6.39). Since the orthogonal projection onto that component is given by $R_-(R_+R_-)^{-1}R_+$, this means that we look for $u_-^0\in\ell^2$, such that $$R_+(\Delta _\phi ^{(0)}-z)u_0+R_+R_-u_-^0=0,$$ i.e. we take \ekv{6.41} { u_-^0=(R_+R_-)^{-1}R_+(z-\Delta _\phi ^{(0)})R_-(R_+R_-)^{-1}v_+=:E_{-+}^0v_+. } If $v_+\in\ell^2({\bf N}_M^\Lambda )$, $0\le M\le N$, then \ekv{6.42} {\cases{ (\Delta _\phi ^{(0)}-z)E_+^0v_++R_-E_{-+}^0v_+= \sum_{0\le \widetilde{M}\le N}h^{{1\over 2}\vert N+1-\widetilde{M}\vert }D_{N+1,\widetilde{M} ;N+1}r_{\widetilde{M}}^*r_{\widetilde{M}} (R_+R_-)^{-1}v_+\cr R_+E_+^0v_+=v_+, }} where $D_{P,M;N+1}$ is defined as in (6.22), with $\widetilde{R}_-^{N,0}$ replaced by the equivalent operator $R_-^{N,0}$. Then (dropping the superscripts in (6.38)) we get \ekv{6.43} { \cases{E_+v_+=E_+^0v_+-\sum_{0\le \widetilde{M}\le N}h^{{1\over 2}\vert N+1-\widetilde{M}\vert }ED_{N+1,\widetilde{M} ;N+1}r_{\widetilde{M}}^*r_{\widetilde{M}} (R_+R_-)^{-1}v_+\cr E_{-+}v_+=E_{-+}^0v_+-\sum_{0\le \widetilde {M}\le N}h^{{1\over 2}\vert N+1-\widetilde{M}\vert }E_-D_{N+1,\widetilde{M} ;N+1}r_{\widetilde{M}}^*r_{\widetilde{M}} (R_+R_-)^{-1}v_+. }} Recall that $\Pi _j$, $j=0,..,N+1$ are the projections associated to the decomposition (6.4) and that $\Pi _j=R_-r_j^*r_j(R_+R_-)^{-1}R_+$, $0\le j\le N$, $\Pi _{N+1}=1-\Pi _{[0,N]}$. Let $A:{\cal H}_{-1}\to {\cal H}_1$. We claim that the following two statements are equivalent: \smallskip \par\noindent (a) $d^{-1}Ad={\cal O}(1):{\cal H}_{-1}\to{\cal H}_1$ for all $(d_j)$ satisfying (6.27). \smallskip \par\noindent (b) $\Pi _jA\Pi _k={\cal O}(h^{{1\over 2}\vert j-k\vert }):{\cal H}_{-1}\to{\cal H}_1$ for all $j,k\in\{ 0,..,N+1\}$. \smallskip \par To see that, we introduce the orthogonal projections $\widehat{\Pi }_j$, $0\le j\le N+1$, with $\widehat{\Pi }_j\widehat{\Pi }_k=0$ for $j\ne k$, $1=\widehat{\Pi }_0+..+\widehat{\Pi }_{N+1}$, by $$\widehat{\Pi }_j=R_-(R_+R_-)^{-{1\over 2}}r_j^*r_j(R_+R_-)^{-{1\over 2}}R_+\hbox{ for }0\le j\le N,\ \widehat{\Pi } _{N+1}=\Pi _{N+1}.$$ Then $d=\sum_0^{N+1}d_j\widehat{\Pi }_j$ and \ekv{*} {d^{-1}Ad=\sum_{j,k=0}^{N+1}{d_k\over d_j}\widehat{\Pi }_jA\widehat{\Pi }_k.} We also notice that $\widehat{\Pi }_j={\cal O}(1)={\cal H}_{\pm 1}\to{\cal H}_{\pm 1}$. We shall show that (a) and (b) are both equivalent to the statement \smallskip \par\noindent (c) $\widehat{\Pi }_jA\widehat{\Pi }_k={\cal O}(h^{{1\over 2}\vert j-k\vert }):{\cal H}_{-1}\to{\cal H}_1$, for all $0\le j,k\le N+1$. \smallskip \par That (c) implies (a) is obvious if we use ($*$), and to get from (a) to (c), it suffices to write $${\cal O}(1)=\widehat{\Pi }_jd^{-1}Ad\widehat{\Pi }_k={d_k\over d_j}\widehat{\Pi }_jA\widehat{\Pi }_k,$$ and choose $d_\nu $ satisfying (6.27) such that $d_k/d_j=h^{-{1\over 2}\vert j-k\vert }$. \par The equivalence between (b) and (c) is an easy consequence of the following estimates $$\Pi _j\widehat{\Pi }_k,\, \widehat{\Pi }_j\Pi _k={\cal O}(h^{{1\over 2}\vert j-k\vert }):{\cal H}_{\pm 1}\to{\cal H}_{\pm 1},$$ that we shall verify: \par When $j=k=N+1$, we have $\Pi _{N+1}\widehat{\Pi }_{N+1}=\widehat{\Pi }_{N+1}\Pi _{N+1}=\Pi _{N+1}=\widehat{\Pi }_{N+1}.$ \par When $j\ne k$ and $N+1\in\{ j,k\}$, then $\widehat{\Pi }_j\Pi _k=\Pi _j\widehat{\Pi }_k=0$. \par For $0\le j,k\le N$, we get $$\Pi _j\widehat{\Pi }_k=R_-r_j^*r_j(R_+R_-)^{-{1\over 2}}r_k^*r_k(R_+R_-)^{-{1\over 2}}R_+$$ and the block matrix element $r_j(R_+R_-)^{-{1\over 2}}r_k^*$ is ${\cal O}(h^{{1\over 2}\vert j-k\vert }):\ell^2\to\ell^2$. Consequently $\Pi _j\widehat{\Pi }_k={\cal O}(h^{{1\over 2}\vert j-k\vert }):{\cal H}_{-1}\to {\cal H}_{1}$. Similarly, $$\widehat{\Pi }_j\Pi _k=R_-(R_+R_-)^{-{1\over 2}}r_j^*r_j(R_+R_-)^{1\over 2}r_k^*r_k(R_+R_-)^{-1}R_+={\cal O}(h^{{1\over 2}\vert j-k\vert }):{\cal H}_{-1}\to{\cal H}_1.$$ Combining (6.24), (6.36) and (6.43), we get: \medskip \par\noindent \bf Proposition 6.2. \it With $E_+^0$, $E_{-+}^0$ given by (6.40), (6.41) and under the assumptions of Proposition 5.1(A), we have \ekv{6.44} { \cases{ \Pi _P(E_+^{N,0}-E_+^0)r_Q^*={\cal O}(1) h^{{1\over 2}(\vert N+1-Q\vert +\vert N+1-P\vert )}:\ell^2\to{\cal H}_1 \cr r_Q(E_-^{N,0}-E_-^{0})\Pi _P={\cal O}(1) h^{{1\over 2}(\vert N+1-Q\vert +\vert N+1-P\vert )}:{\cal H}_{-1}\to \ell^2 \cr r_{\widetilde{P}}(E_{-+}^{N,0}-E_{-+}^0) r_Q^* ={\cal O}(1)h^{{1\over 2}(\vert N+1-Q\vert +\vert N+1-\widetilde{P}\vert )}:\ell^2\to\ell^2, } } for $0\le P\le N+1$, $0\le \widetilde{P},Q\le N$, where $E_-^0:=(E_+^0)^*$.\rm\medskip \par Here the second equation in (6.44) is obtained by duality, using that $\Pi _{N+1}$ is the orthogonal projection (6.1) and that $\Pi _P$ is given after (6.5). \par Now let $M>N$ and let us compare $${\cal P}^{N,0}, {\cal P}^{M,0}$$ and their inverses for $z$ in the domain of wellposedness of the "smaller" problem ${\cal P}^{N,0}$. Let $r=r_{[0,N]}$ denote the natural restriction operator: $\ell^2({\bf N}^\Lambda _{[0,M]})\to\ell^2({\bf N}^\Lambda _{[0,N]})$, and notice that \ekv{6.45} { R_+^{N,0}=r_{[0,N]}R_+^{M,0},\ R_-^{N,0}=R_-^{M,0}r_{[0,N]}^*, } where $r_{[0,N]}^*$ is the adjoint: $\ell^2({\bf N}^\Lambda _{[0,N]})\to \ell^2({\bf N}^\Lambda _{[0,N]})$. To shorten the notations, we write ${\cal P}={\cal P}^{N,0}$, $\widetilde{{\cal P}}={\cal P}^{M,0}$, and similarly for the associated quantities. In order to solve Gr($N$,0): \ekv{6.46} { \cases{ (\Delta _\phi ^{(0)}-z)u+R_-u_-=v\cr R_+u=v_+, } } we consider the bigger problem Gr($M$,0) \ekv{6.47} {\cases{ (\Delta _\phi ^{(0)}-z)u+\widetilde{R}_-\widetilde{u}_- =v\cr \widetilde{R}_+u=\widetilde{v}_+, }} and write the solution as \ekv{6.48} {\cases{ u=\widetilde{E}v+\widetilde{E}_ +\widetilde{v}_+\cr \widetilde{u}_-=\widetilde{E}_-v+ \widetilde{E}_{-+}\widetilde{v}_+. }} We want (6.46) to be fulfilled, so we get the condition \ekv{6.49} { R_-u_-=\widetilde{R}_-\widetilde{u}_-. } The necessary and sufficient condition on $\widetilde{u}_-$ for (6.49) to have a solution $u_-$ is \ekv{6.50} { r_{[N+1,M]}\widetilde{u}_-=0,} where $r_{[N+1,M]}:\ell^2({\bf N}^\Lambda _{[0,M]})\to\ell^2({\bf N}^\Lambda _{[N+1,M]})$ is the restriction operator, and the corresponding $u_-$ is then \ekv{6.51} { u_-=r_{[0,N]}\widetilde{u}_-. } We then get a solution of (6.46) iff \ekv{6.52} { r_{[N+1,M]}\widetilde{E}_-v+r_{[N+1,M]} \widetilde{E}_{-+}\widetilde{v}_+=0, } \ekv{6.53} { r_{[0,N]}\widetilde{v}_+=v_+. } (6.52) is equivalent to \ekv{6.54} { \widetilde{E}_{-+} \widetilde{v}_++r^*_{[0,N]}w_-=-\widetilde{E} _-v,\hbox{ for some }w_-\in\ell^2({\bf N}^\Lambda _{[0,N]}). } (6.54), (6.53) lead to a new Grushin problem, namely to invert the matrix \ekv{6.55} { \widetilde{{\cal E}}_{-+}:=\pmatrix{\widetilde{E}_{-+} &r^*_{[0,N]}\cr r_{[0,N]}&0}, } which is wellposed in the range of wellposedness of ${\cal P}$ given in Proposition 5.1(A). This follows from Remark 5.5 and the fact that $$\widetilde{E}_{-+}=(\widetilde{R}_+ \widetilde{R}_-)^{-1} \widetilde{R}_+ (z-\Delta _\phi ^{(0)})\widetilde{R}_-( \widetilde{R}_+\widetilde{R}_-)^{-1}+ {\cal O}(h^{1\over 2}),$$ by Proposition 6.2. Modulo ${\cal O}(h^{1\over 2})$ we obtain a block diagonal matrix and the diagonal block at $(j,j)$ with $0\le j\le M$ is given by $$(\phi ''(0)^{-{1\over 2}}\otimes ..\otimes \phi ''(0)^{-{1\over 2}})(z-(\phi ''(0)\otimes 1\otimes ..\otimes 1+...1\otimes ..\otimes 1\otimes \phi ''(0)))(\phi ''(0)^{-{1\over 2}}\otimes ..\otimes \phi ''(0)^{-{1\over 2}}).$$ Here the tensor products are of length $j$ and for $j=0$ the expression above should be replaced by $z$. \par Let \ekv{6.56} {\pmatrix{F&F_+\cr F_-&F_{-+}}} be the inverse of (6.55), so that \ekv{6.57} { \pmatrix{\widetilde{v}_+\cr w_-}=\pmatrix{F &F_+\cr F_-&F_{-+}}\pmatrix{-\widetilde{E}_-v\cr v_+}, } i.e. \ekv{6.58} {\widetilde{v}_+=-F\widetilde{E}_-v+F_+v_+,\ w_-=-F_-\widetilde{E}_-v+F_{-+}v_+.} The solution of (6.46) is then given from (6.48), (6.51), (6.58): $$\eqalign{ u&=\widetilde{E}v+\widetilde{E}_+(-F \widetilde{E}_-v+F_+v_+)\cr u_-&=r_{[0,N]}(\widetilde{E}_-v+\widetilde{E}_{-+}(-F \widetilde{E}_-v+F_+v_+)), }$$ i.e. \ekv{6.59} {\cases{ u=(\widetilde{E}- \widetilde{E}_+ F \widetilde{E}_-) v +\widetilde{E}_+ F_+v_+ \cr u_-=r_{[0,N]}(\widetilde{E}_--\widetilde{E}_ {-+} F \widetilde{E}_- )v+ r_{[0,N]} \widetilde{E}_{-+} F_+v_+. }} This can be further simplified, if we use the identity $\widetilde{E}_{-+}F+r^*F_-=1$, with $r=r_{[0,N]}$ for short. Then $$(\widetilde{E}_--\widetilde{E}_{-+}F \widetilde{E}_-)= (1-\widetilde{E}_{-+} F)\widetilde{E}_-= r^*F_-\widetilde{E}_-,$$ and since $rr^*=1$, we get \ekv{6.60} {\cases{ u=(\widetilde{E}-\widetilde{E}_+ F\widetilde{E}_-)v+ \widetilde{E}_+ F_+ v_+ \cr u_-=F_-\widetilde{E}_-v+r_{[0,N]} \widetilde{E}_{-+}F_+v_+. }} We next use the relation $\widetilde{E}_{-+} F_++r^*F_{-+}=0$, to rewrite the last equation as $$u_-=F_-\widetilde{E}_-v-rr^*F_{-+}v_+,$$ and using again that $rr^*=1$, we get the solution of the small problem ${\cal P}={\cal P}^{N,0}$ as \ekv{6.61} {\cases{ u=Ev+E_+v_+\cr u_-=E_-v+E_{-+}v_+ } \hbox{ with } \cases{ E=\widetilde{E}-\widetilde{E}_+F \widetilde{E}_-,\ E_+=\widetilde{E}_+F_+\cr E_-=F_-\widetilde{E}_-,\ E_{-+}=-F_{-+} }. } \par The next goal is to get asymptotics for $F_{-+}$, $F_-$ similar to (6.44) with $N$ replaced by $M$. Define $\widetilde{E}_+^0$ and $\widetilde{E}_{-+}^0$ as in (6.40) (6.41): \ekv{6.62} {\cases{\widetilde{E}_+^0=\widetilde{R}_- (\widetilde{R}_+ \widetilde{R}_-)^{-1} \cr\widetilde{E}_{-+}^0=( \widetilde{R}_+\widetilde{R}_-)^{-1} \widetilde{R}_+(z-\Delta _\phi ^{(0)})\widetilde{R}_-(\widetilde{R}_ +\widetilde{R}_-)^{-1},}} so that analogously to (6.44) \ekv{6.63} {\cases{ \Pi _P(\widetilde{E}_+-\widetilde{E}_+^0) r_Q^*={\cal O}(1) h^{{1\over 2}(\vert M+1-Q\vert +\vert M+1-P\vert )}:\ell^2\to{\cal H}_1 \cr r_Q(\widetilde{E}_--\widetilde{E}_-^0)\Pi _P={\cal O}(1) h^{{1\over 2}(\vert M+1-Q\vert +\vert M+1-P\vert )}:{\cal H}_{-1}\to \ell^2 \cr r_{\widetilde{P}}(\widetilde{E}_{-+} -\widetilde{E}_{-+}^0) r_Q^* ={\cal O}(1)h^{{1\over 2}(\vert M+1-Q\vert +\vert M+1-\widetilde{P}\vert )}:\ell^2\to\ell^2,} } for $0\le P\le M+1$, $0\le Q,\widetilde{P}\le M$, where $\widetilde{E}_-^0$ is defined to be the adjoint of $\widetilde{E}_+^0$. \par Let \ekv{6.64} { \pmatrix{F^0&F_+^0\cr F_-^0&F_{-+}^0}={\pmatrix{ \widetilde{E}_{-+}^0&r_{[0,N]}^*\cr r_{[0,N]}&0}}^{-1}=:(\widetilde{{\cal E}}_{-+}^0)^{-1}. } Let $\lambda _j>0$, $0\le j\le M+1$ and let $\lambda _{[0,M]}={\rm diag\,}(\lambda _j)_{0\le j\le M}$, $\lambda _{[0,N]}={\rm diag\,}(\lambda _j)_{0\le j\le N}$. Let \ekv{6.65}{\Lambda =\pmatrix{\lambda _{[0,M]}&0\cr 0&\lambda _{[0,N]}},} viewed as an operator on $$(\oplus_0^M\ell^2({\bf N}_j^\Lambda ))\times (\oplus_0^N\ell^2({\bf N}_j^\Lambda )).$$ As before we define the action of $\lambda $ on ${\cal H}_{\pm 1}$ and on $L^2({\bf R}^\Lambda )$. We also need a second system of weights $\mu _j$ and define $\mu _{[0,N]}$, $\mu _{[0,M]}$ and ${\cal M} =\pmatrix{\mu _{[0,M]}&0\cr 0&\mu _{[0,N]}}$ analogously. (6.63) can be reformulated as \ekv{6.66} {\cases{\lambda (\widetilde{E}_+-\widetilde{E}_+^0)\mu _{[0,M]}^{-1}={\cal O}(1):\ell^2\to{\cal H}_1, \cr \mu _{[0,M]}(\widetilde{E}_-- \widetilde{E}_-^0) \lambda ^{-1}={\cal O}(1): {\cal H}_{-1}\to\ell^2, \cr\lambda _{[0,M]}(\widetilde{E}_{-+}- \widetilde{E}_{-+}^0)\mu _{[0,M]}^{-1}={\cal O}(1):\ell^2\to\ell^2,}} for all $\lambda $, $\mu $ as above with \ekv{6.67} { {\lambda _{j+1}\over \lambda _j}, {\mu _{j+1}\over\mu _j}\in [h^{1\over 2},h^{-{1\over 2}}],\ 0\le j\le M, } \ekv{6.68} {\lambda _{M+1}=\mu _{M+1}.} It follows from (6.66) that \ekv{6.69} {\Lambda (\widetilde{{\cal E}}_{-+}-\widetilde{{\cal E}}^0_{-+}){\cal M}^{-1}={\cal O}(1),} for all $\mu $, $\lambda $ which satisfy (6.67,68). We also have \ekv{6.70} {\Lambda \widetilde{{\cal E}}_{-+}\Lambda ^{-1}={\cal O}(1)} and similarly with $\Lambda $ replaced by one of $\Lambda ^{-1}$, ${\cal M}^{\pm 1}$ and/or $\widetilde{{\cal E}}_{-+}$ replaced by $(\widetilde{{\cal E}}^0_{-+})^{\pm 1}$, $(\widetilde{{\cal E}}_{-+})^{-1}$. Then \ekv{6.71} {\Lambda (\widetilde{{\cal E}}_{-+}^{-1}-(\widetilde{{\cal E}}^0_{-+})^{-1}){\cal M}^{-1}=\Lambda \widetilde{{\cal E}}_{-+}^{-1}\Lambda ^{-1}\Lambda (\widetilde{{\cal E}}_{-+}^0-\widetilde{{\cal E}}_{-+}){\cal M}^{-1}{\cal M}(\widetilde{{\cal E }}_{-+}^0)^{-1}{\cal M}^{-1}={\cal O}(1),} for all $\mu $, $\lambda $ satisfying (6.67), (6.68). Equivalently, if we introduce the block matrix notation $A_{j,k}=r_jAr_k^*$, then $$\eqalignno{ (F-F^0)_{j,k}&={\cal O}(h^{{1\over 2}(\vert M+1-j\vert +\vert M+1-k\vert )}),\ 0\le j,k\le M,&(6.72)\cr (F_+-F_+^0)_{j,k}&= {\cal O}(h^{{1\over 2}(\vert M+1-j\vert +\vert M+1-k\vert )}),\ 0\le j\le M,\ 0\le k\le N,\cr (F_--F_-^0)_{j,k}&={\cal O}(h^{{1\over 2}(\vert M+1-j\vert +\vert M+1-k\vert )}),\ 0\le j\le N,\ 0\le k\le M,\cr (F_{-+}-F_{-+}^0)_{j,k}&={\cal O}(h^{{1\over 2}(\vert M+1-j\vert +\vert M+1-k\vert )}),\ 0\le j,k\le N. }$$ \par For $\widetilde{E}_{-+}^0$ we have a complete asymptotic expansion \ekv{6.73}{\widetilde{E}^0_{-+;j,k}\sim h^{{1\over 2}\vert j-k\vert }\sum_{\nu =0}^\infty h^\nu A_{j,k}^\nu \hbox{ in }{\cal L}(\ell^2,\ell^2),} and it follows that the inverse (cf. (6.64)) of the corresponding Grushin problem $\widetilde{{\cal E}}_{-+}^0$ has the same structure. $$\eqalignno{F_{j,k}^0&\sim h^{{1\over 2}\vert j-k\vert }\sum_{\nu =0}^\infty h^\nu B_{j,k}^{0,\nu },\ 0\le j,k\le M,&(6.74)\cr F_{+;j,k}^0&\sim h^{{1\over 2}\vert j-k\vert }\sum_{\nu =0}^\infty h^\nu B_{+;j,k}^{0,\nu },\ 0\le j\le M,\ 0\le k\le N, \cr F_{-;j,k}^0&\sim h^{{1\over 2}\vert j-k\vert }\sum_{\nu =0}^\infty h^\nu B_{-;j,k}^{0,\nu },\ 0\le j\le N,\ 0\le k\le M, \cr F_{-+;j,k}^0&\sim h^{{1\over 2}\vert j-k\vert }\sum_{\nu =0}^\infty h^\nu B_{-+;j,k}^{0,\nu },\ 0\le j,k\le N . }$$ Combining this with (6.72) and letting $M\to \infty $ we obtain a complete asymptotic expansion for $F_{-+}=-E_{-+}^{N,0}$: \ekv{6.75} { -F_{-+;j,k}\sim h^{{1\over 2}\vert j-k\vert }\sum_{\nu =0}^\infty h^\nu B^\nu _{-+;j,k},\ 0\le j,k\le N. } For $F_{\pm}$ we only have a partial asymptotics with the limitations of (6.72) but again it will be advantageous to let $M\to\infty $. \par We now look at $E_+$. The first equation in (6.63) tells us that $$(\widetilde{E}_+-\widetilde{E}_+^0)r_Q^*= {\cal O}(1)h^{{1\over 2}\vert M+1-Q\vert }:\ell^2\to {\cal H}_1.$$ Write $$E_+=\widetilde{E}_+F_+=(\widetilde{E}_+- \widetilde{E}_+^0)F_++\widetilde{E}_+^0( F_+-F_+^0)+ \widetilde{E}_+^0 F_+^0.$$ Here $$(\widetilde{E}_+-\widetilde{E}_+^0 )F_+= {\cal O}(1) \sum_{Q=0}^Mh^{{1\over 2} (\vert M+1-Q\vert +(Q-N)_+) }:\ell^2\to{\cal H}_1,$$ so that $(\widetilde{E}_+-\widetilde{E}_+^0 )F_+={\cal O}(1)h^{{1\over 2}\vert M+1-N\vert }:\ell^2\to{\cal H}_1$. We also have $$\widetilde{E}_+^0(F_+-F_+^0)={\cal O}(1)\sum_{Q=0}^M h^{{1\over 2}(\vert M+1-Q\vert +\vert M+1-N\vert )} ={\cal O}(1)h^{{1\over 2}\vert M+1-N\vert }:\ell^2\to{\cal H}_1.$$ Consequently, $$E_+^{N,0}=\widetilde{E}_+^0F_+^0+{\cal O} (1)h^{{1\over 2}(M+1-N)}:\ell^2 \to {\cal H}_1.$$ Here $F_+^0$ has a complete asymptotic expansion given by (6.74) and $\widetilde{E}_+^0$ is given by (6.62): $$E_+^{N,0}=R_-^{M,0}(R_+^{M,0}R_-^{M,0} )^{-1} F_+^0+{\cal O}(1)h^{{1\over 2}(M+1-N)}:\ell^2\to{\cal H}_1,$$ where $(R_+^{M,0}R_-^{M,0})^{-1}$ has a complete asymptotic expansion of the same type as $R_+^{M,0}R_-^{M,0}$ (c.f. (5.29)), so we conclude that for every $M\ge N$: \ekv{6.76} {E_+^{N,0}=R_-^{M,0}C^M+{\cal O}(1)h^{{1\over 2}(M+1-N)}:\ell^2\to {\cal H}_1,} where \ekv{6.77} { C_{j,k}^M\sim \sum_{\nu =0}^\infty h^{{1\over 2}\vert j-k\vert +\nu }D_{j,k}^{M,\nu },\hbox{ in }{\cal L}(\ell^2,\ell^2), } for $0\le j\le M$, $0\le k\le N$. Summing up, we have \medskip \par\noindent \bf Proposition 6.3. \it $E_{-+}^{N,0}$ has a complete asymptotic expansion in ${\cal L}(\ell^2,\ell^2)$, that can be written at the level of block matrix elements: \ekv{6.78} { E_{-+;j,k}^{N,0}\sim h^{{1\over 2}\vert j-k\vert }\sum_{\nu =0}^\infty h^\nu B^\nu _{-+;j,k},\ 0\le j,k\le N. } For every $M\ge N$, we have (6.76,77) for $E_+^{N,0}$.\rm\medskip \par Using Proposition 6.2, (6.40), (6.41), we also get the leading terms in the asymptotic expansions (6.78), (6.77): \eekv{6.79} {B^0_{-+;j,j}=(\phi ''(0)\otimes ..\otimes \phi ''(0))^{-{1\over 2}}}{\hskip 2cm (z-(\phi ''(0)\otimes 1\otimes ..\otimes 1+..+1\otimes ..\otimes 1\otimes \phi ''(0)))(\phi ''(0)\otimes ..\otimes \phi ''(0))^{-{1\over 2}},} \ekv{6.80} {D_{j,j}^{M,0}=(\phi ''(0)\otimes ..\otimes \phi ''(0))^{-1},\ 0\le j\le M.} \par We now want to do the same job with Gr($N$,1) as we did with Gr($N$,0), and the only slightly new thing is to analyze the block matrix of $\phi ''(x)$, viewed as an operator \eekv{6.81} {\ell^2(\Lambda )\otimes ({\cal L}_0\oplus ..\oplus{\cal L}_N\oplus({\cal H}_1\cap{\cal L}_{[0,N]}^\perp))\to} {\ell^2(\Lambda )\otimes ({\cal L}_0\oplus ..\oplus{\cal L}_N\oplus({\cal H}_{-1}\cap{\cal L}_{[0,N]}^\perp)).} According to (3.12) (or by reviewing more directly the arguments leading to that result), if ${\cal M}=\{ 1,..,M\}$, for $1\le M\le N$, or ${\cal M}=\emptyset $, then for $\nu ,\mu \in\Lambda $, $m\in\Lambda ^{{\cal M}}$, $\phi ''_{\nu ,\mu}(x)Z^*_{m\vert {\cal M}}(e^{-\phi /2h})$ is a finite sum of terms of the type \ekv{6.82} {\sum_{\ell\in\Lambda ^L}h^XZ^*_{m\vert M_0}Z^*_{\ell\vert L}(\Phi _{\nu ,\mu ,m\vert M_1,\ell\vert L}(x)e^{-\phi /2h}),} where ${\cal M}=M_0\cup M_1$ is a partition and $L$ is a finite set, possibly with $M_1$ or $L$ empty. $\Phi $ is a standard tensor and \ekv{6.83} {X={1\over 2}(\# M_1+\# L)+\widetilde{N},\ {\bf N}\ni\widetilde{N}\le N_1\in{\bf N},} where $N_1\in {\bf N}$ is any fixed number. Moreover $\Phi $ is independent of $x$, when $$\cases{\# M_0+\# L0$ sufficiently small and uniformly for all $\rho \in W_a$, we have the asymptotic expansion in ${\cal L}(\ell^2\otimes \ell^2,\ell^2\otimes \ell^2)$, that can be written for the block diagonal elements: \ekv{7.23} { (\rho \otimes 1)^{-1}E_{-+;j,k}^{N,1}(\rho \otimes 1)\sim h^{{1\over 2}\vert j-k\vert} \sum_{\nu =0}^\infty h^\nu (\rho \otimes 1)^{-1}B_{-+;j,k}^\nu (\rho \otimes 1),\ 0\le j,k\le N. } Here $B_{-+;j,k}^\nu $ are the same as in (6.99). \par For $M\ge N$, we have: \ekv{7.24} { (\rho \otimes 1)^{-1}E_+^{N,1}(\rho \otimes 1)=R_-^{M,1}(\rho \otimes 1)^{-1}C^M(\rho \otimes 1)+{\cal O}(1)h^{{1\over 2}(M+1-N)}:\ell^2\otimes \ell^2\to \ell^2\otimes {\cal H}_1. } Here $C^M$ is the same as in Proposition 6.5, and we have the asymptotic expansion for the block matrix elements, valid uniformly with respect to $\rho\in W_a $: \ekv{7.25} { (\rho \otimes 1)^{-1}C_{j,k}^M(\rho \otimes 1)\sim \sum_{\nu =0}^\infty h^{{1\over 2}\vert j-k\vert +\nu }(\rho \otimes 1)^{-1}D_{j,k}^{M,\nu }(\rho \otimes 1), } for $0\le j\le M$, $0\le k\le N$.\rm\medskip %\vfill\eject \bigskip \centerline{\bf 8. Parameter dependent exponents.} \medskip \par In this section we carry out an essential preparation for controling the thermodynamical limit of the correlations. For that, we need to estimate the variation of the correlations, when the exponent $\phi =\phi _t(x)$ depends on a parameter $t\in [0,1]$: \ekv{8.1{\rm H}} {\phi _t(x)=\phi _{t,0}(x)+C(t;h),} with $C(t;h)$ independent of $x$ and with $\phi _{t,0}$ independent of $h$. We assume that $\phi _t(x)$ is of class $C^1$ in $t$ and smooth in $x\in{\bf R}^\Lambda $. Further assumptions will be given later on. We assume that $C(t;h)$ is chosen so that \ekv{8.2{\rm H}}{\int e^{-\phi _t(x)/h}dx=1.} \par We start the section by making some formal computations. After that we will introduce some precise assumptions on $\phi _t$ that justify the formal computations. Finally we will estimate the various terms that we get. The estimates will be summed up in Proposition 8.4. \par We are interested in \ekv{8.3} {{\rm Cor}_t(u,v)=\int e^{-\phi _t(x)/h}(u-\langle u\rangle _t)(v-\langle v\rangle _t)dx,} where $\langle u\rangle _t$ denotes the expectation of $u$ with respect to $e^{-\phi _t/h}dx$, and where $u$ and $v$ are supposed to be independent of $t$. Since $u-\langle u\rangle _t$ and $v-\langle v_t\rangle $ have expectation 0, we get \ekv{8.4} {-\partial _t{\rm Cor}_t(u,v)=\int e^{-\phi _t(x)/h}{\partial _t\phi _t(x)\over h}(u-\langle u\rangle _t)(v-\langle v\rangle _t)dx.} \par Assume that \ekv{8.5{\rm H}} {\partial _t\phi _{t,0}(0)=0,\ \partial _x\partial _t\phi _{t,0}(0)=0.} Then \ekv{8.6} {\partial _t\phi _{t,0}(x)=\sum_{j,k}\widetilde{\Phi }_{j,k}(x)x_jx_k=\langle \widetilde{\Phi }(x)x,x\rangle ,} with $\widetilde{\Phi }=\widetilde{\Phi }_t$ given by \ekv{8.7} {\widetilde{\Phi }_{j,k}(x)=\int_0^1(1-s)\partial _{x_j}\partial _{x_k}\partial _t\phi _{t,0}(sx)ds.} Now assume that \ekv{8.8{\rm H}} {\partial _x\phi _t(x)=A_t(x)x,} where the matrix $A_t(x)$ is $C^1$ in $t$, $C^\infty $ in $x$ and invertible. Combining this with (8.6), we get \ekv{8.9} {\partial _t\phi _{t,0}(x)=\sum_{j,k}\Phi _{j,k}(x)\partial _{x_j}\phi \partial _{x_k}\phi =\langle \Phi (x)\partial _x\phi (x),\partial _x\phi (x)\rangle ,} with \ekv{8.10} {\Phi (x)={^t\hskip -1pt A}(x)^{-1}\widetilde{\Phi }(x)A(x)^{-1}.} Here and in the following, we often drop the subscript $t$. Later on it will be useful to keep in mind that $\Phi _{j,k}(x)$ is symmetric. \par With $\phi =\phi _t$, we define $Z_j,Z_j^*$ as in section 1. A straight forward computation shows that \ekv{8.11} { e^{-\phi /2h}{\partial _t\phi \over h}=\sum_{j,k}Z_j^*Z_k^*(e^{-\phi /2h}\Phi _{j,k})+h^{1\over 2}\sum_j Z_j^*(e^{-\phi /2h}\Psi _j)+D(x;h)e^{-\phi /2h}, } where \ekv{8.12} { \Psi _j=2\sum_k\partial _{x_k}\Phi _{j,k}, } \ekv{8.13} { D=\sum_{j,k}(\partial _{x_j}\partial_{x_k}\phi )\Phi _{j,k}+h\sum_{j,k}\partial _{x_j}\partial _{x_k}\Phi _{j,k}+{\partial _tC(t;h)\over h}. } Using this in (8.4), we get \ekv{8.14} { -\partial _t{\rm Cor}_t(u,v)={\rm I}+{\rm II}+{\rm III}, } where \ekv{8.15} { \cases{{\rm I}=\int\sum_{j,k}Z_j^*Z_k^*(e^{-\phi /2h}\Phi _{j,k})e^{-\phi /2h}(u-\langle u\rangle )(v-\langle v\rangle )dx, \cr {\rm II}=\int h^{1\over 2}\sum_j Z_j^*(e^{-\phi /2h}\Psi _j)e^{-\phi /2h}(u-\langle u\rangle )(v-\langle v\rangle )dx,\cr {\rm III}=\int e^{-\phi /2h}De^{-\phi /2h}(u-\langle u\rangle )(v-\langle v\rangle )dx. } } Since $Z_j\circ e^{-\phi /2h}=e^{-\phi /2h}h^{1/2}\partial _{x_j}$, we get after an integration by parts, \eeekv{8.16} { {\rm I}=\int\sum_{j,k}e^{-\phi /2h}\Phi _{j,k}Z_jZ_k(e^{-\phi /2h}(u-\langle u\rangle )(v-\langle v\rangle ))dx } { =h\int \sum_{j,k}e^{-\phi /2h}\Phi _{j,k}e^{-\phi /2h}\partial _{x_j}\partial _{x_k}((u-\langle u\rangle )(v-\langle v\rangle ))dx } { ={\rm I}_1+{\rm I}_2+{\rm I}_3, } where \ekv{8.17} { {\rm I}_1=2h\int \sum_{j,k}\Phi _{j,k}(e^{-\phi /2h}\partial _{x_j}u)(e^{-\phi /2h}\partial _{x_k}v) dx, } and $$\cases{ {\rm I}_2=h\int\sum_{j,k}e^{-\phi /2h}\Phi _{j,k}(\partial _{x_j}\partial _{x_k}u)e^{-\phi /2h}(v-\langle v\rangle )dx,\cr {\rm I}_3=h\int\sum_{j,k}e^{-\phi /2h}\Phi _{j,k}(\partial _{x_j}\partial _{x_k}v)e^{-\phi /2h}(u-\langle u\rangle )dx.} $$ Here we used the symmetry of $(\Phi _{j,k})$ to get ${\rm I}_1$. We need to transform ${\rm I}_2$, ${\rm I}_3$ further. \par Later in this section we shall solve \ekv{8.18} {\cases{ e^{-\phi /2h}(u-\langle u\rangle )=\sum_\nu Z_\nu ^*(\widetilde{u}_\nu )\,\,(=d_\phi ^*(\sum \widetilde{u}_\nu dx_\nu )),\cr e^{-\phi /2h}(v-\langle v\rangle )=\sum_\nu Z_\nu ^*(\widetilde{v}_\nu ),\cr e^{-\phi /2h}D=\sum Z_\nu ^*(\widetilde{D}_\nu ). }} As for the last equation, we see formally that \ekv{8.19} { \langle D\rangle =0. } This follows from (8.11), since $$\langle {\partial _t\phi \over h}\rangle =-\partial _t\int e^{-\phi _t/h}dx=-\partial _t(1)=0,$$ and $$\int e^{-\phi /2h}Z_j^*wdx=\int Z_j(e^{-\phi /2h})w dx=0,$$ unders suitable assumptions on $w$ which will be verified. \par Substitution of the second equation of (8.18) into the expression for ${\rm I}_2$ and integration by parts gives \ekv{8.20} { {\rm I}_2=h^{3\over 2}\int\sum_{j,k,\nu }e^{-\phi /2h}\partial _{x_\nu }(\Phi _{j,k}\partial _{x_j}\partial _{x_k}u)\widetilde{v}_\nu dx={\rm I}_{2,1}+{\rm I}_{2,2}, } where \ekv{8.21} {\cases{ {\rm I}_{2,1}=h^{3\over 2}\int\sum_{j,k,\nu }e^{-\phi /2h}(\partial _{x_\nu }\Phi _{j,k})(\partial _{x_j}\partial _{x_k}u)\widetilde{v}_\nu dx,\cr {\rm I}_{2,2}=h^{3\over 2}\int\sum_{j,k,\nu }e^{-\phi /2h}\Phi _{j,k}(\partial _{x_\nu }\partial _{x_j}\partial _{x_k}u)\widetilde{v}_\nu dx. }} Since ${\rm I}_3$ is obtained from ${\rm I}_2$ by exchanging $u$ and $v$, we get ${\rm I}_3={\rm I}_{3,1}+{\rm I}_{3,2}$, with ${\rm I}_{3,1}$, ${\rm I}_{3,2}$ as in (8.21), with $u$ replaced by $v$ and $\widetilde{v}_\nu $ by $\widetilde{u}_\nu $. \par After an integration by parts and application of (8.18), we get \ekv{8.22} { {\rm II}={\rm II}_1+{\rm II}_2, } \ekv{8.23} {\cases{ {\rm II}_1=h\int\sum_{j,\nu }\Psi _j(e^{-\phi /2h}\partial _{x_j}u)Z_\nu ^*\widetilde{v}_\nu dx,\cr {\rm II}_2=h\int\sum_{j,\nu }\Psi _j(e^{-\phi /2h}\partial _{x_j}v)Z_\nu ^*\widetilde{u}_\nu dx. }} We observe that ${\rm II}_1$ and ${\rm II}_2$ differ only by a permutation of $u$ and $v$ and their related quantities. By integration by parts, we get \ekv{8.24} { {\rm II}_1={\rm II}_{1,1}+{\rm II}_{1,2}, } \ekv{8.25} {\cases{ {\rm II}_{1,1}=h^{3\over 2}\int\sum_{j,\nu }(\partial _{x_\nu }\Psi _j)(e^{-\phi /2h}\partial _{x_j}u)\widetilde{v}_\nu dx,\cr {\rm II}_{1,2}=h^{3\over 2}\int\sum_{j,\nu }\Psi _j(e^{-\phi /2h}\partial _{x_\nu }\partial _{x_j}u)\widetilde{v}_\nu dx. }} Similarly, we have ${\rm II}_2={\rm II}_{2,1}+{\rm II}_{2,2}$, where ${\rm II}_{2,i}$ is obtained from ${\rm II}_{1,i}$, by replacing $u$ by $v$ and $\widetilde{v}_\nu $ by $\widetilde{u}_\nu $. \par Next consider ${\rm III}$ in (8.15). Using (8.18) and an integration by parts, we get \ekv{8.26} { {\rm III}={\rm III}_1+{\rm III}_2, } \ekv{8.27} {\cases{ {\rm III}_1=h^{1\over 2}\int\sum_\nu \widetilde{D}_\nu (\partial _{x_\nu }u)e^{-\phi /2h}(v-\langle v\rangle )dx,\cr {\rm III}_2=h^{1\over 2}\int\sum_\nu \widetilde{D}_\nu (\partial _{x_\nu }v)e^{-\phi /2h}(u-\langle u\rangle )dx. }} Again the two terms are analogous. Applying (8.18) and integrating by parts, we get \ekv{8.28} { {\rm III}_1={\rm III}_{1,1}+{\rm III}_{1,2}, } \ekv{8.29} {\cases{ {\rm III}_{1,1}=h^{1\over 2}\int\sum_{\nu ,\mu }\widetilde{v}_\mu (Z_\mu \widetilde{D}_\nu )(\partial _{x_\nu }u)dx,\cr {\rm III}_{1,2}=h\int\sum_{\nu ,\mu }\widetilde{v}_\mu \widetilde{D}_\nu (\partial _{x_\mu }\partial _{x_\nu }u)dx. }} Clearly ${\rm III}_2={\rm III}_{2,1}+{\rm III}_{2,2}$, where ${\rm III}_{2,i}$ is obtained from ${\rm III}_{1,i}$ by replacing $u$ by $v$ and $\widetilde{v}_\mu $ by $\widetilde{u}_\mu $. This completes our formal calculations: \eeekv{8.30} {-\partial _t{\rm Cor\,}(u,v)={\rm I}_{1}+{\rm I}_{2,1}+{\rm I}_{2,2}+{\rm I}_{3,1}+{\rm I}_{3,2}+} {\hskip 3cm {\rm II}_{1,1}+{\rm II}_{1,2}+{\rm II}_{2,1}+{\rm II}_{2,2}+} {\hskip 4cm {\rm III}_{1,1}+{\rm III}_{1,2}+{\rm III}_{2,1}+{\rm III}_{2,2}.} \par Let $W=W_\Lambda $ be a set of positive weight functions $\rho :\Lambda \to ]0,\infty [$, with $1\in W$ and such that $\rho \in W\Rightarrow 1/\rho \in W$. First of all we assume that $\phi =\phi _t$ satisfies the assumptions of the earlier sections uniformly w.r.t. $t$. (Actually in the next sections we shall see that the sets of weights we use in this section are smaller than the corresponding sets of weights in sections 7.) More precisely we assume that ($\widetilde{{\rm H}1}$) (section 7) holds uniformly in $t$. We assume (H2) and we assume that (H3) holds uniformly in $t$. We strengthen (H4) to: \eeeekv{\widetilde{{\rm H}4}} {\phi _t'(x)=A_t(x)x\hbox{, where }\rho ^{-1}A_t(x)\rho \hbox{ is 2 standard and }\rho (\ell )(\partial _{x_\ell}A_t)\circ \rho ^{-1}, } {\rho ^{-1}\circ \rho (\ell )\partial _{x_\ell} A_t\hbox{ are 3 standard uniformly for }\rho \in W,\ t\in [0,1].\hbox{ Moreover, }A_t(x)\hbox{ has} } {\hbox{an inverse }B_t(x)\hbox{ such that }\rho ^{-1}B_t(x)\rho ={\cal O}(1):\ell^p\to\ell^p,\ 1\le p\le \infty ,}{\hbox{uniformly for }0\le t\le 1,\ \rho \in W.} It follows that $\rho ^{-1}A_t^{-1}(x)\rho $ is 2 standard and that $\rho (\ell )(\partial _{x_\ell}A_t^{-1})\circ \rho ^{-1}$, $\rho ^{-1}\circ \rho (\ell )\partial _{x_\ell} A_t^{-1}$ are 3 standard, uniformly for $0\le t\le 1$, $\rho \in W$. The most natural choice of $A_t$ seems to be $A_t=\int_0^1 \phi _t''(sx)ds$. With that choice we only have to check the statement about the inverse of $A_t$, since the other properties follow from the previous assumptions on $\phi $. Let $W_a\subset W$ be a set of weights as with $\rho \in W_a\Rightarrow 1/\rho \in W_a$ such that (7.2) holds, with $\phi =\phi _t$. We let $a$ be fixed with \ekv{8.32} { 00$. Then for the solution in $\ell^2\otimes {\cal H}_1$ of the last equation in (8.18), we have \ekv{8.41} {\sum_{\nu }\Vert \rho _1(\nu )\widetilde{D}_\nu \Vert ^2+\sum_\nu \sum_\mu \Vert \rho _1(\nu )Z_\mu \widetilde{D}_\nu \Vert ^2\le {\cal O}(1)h.} \par The justification of our derivation of (8.30) is now immediate, and next we shall estimate the various terms that appear in that equation. \smallskip \par\noindent \it Estimate of $I_1$ (see (8.17)). \rm We rewrite the integrand in (8.17) as \ekv{8.42} { e^{-\phi /h}\sum_{j,k}(\rho _0(j)\rho _0(k)\Phi _{j,k}(x)){1\over \rho _u(j)\rho _0(j)}(\rho _u(j)\partial _{x_j}u){1\over \rho _v(k)\rho _0(k)}(\rho _v(k)\partial _{x_k}v). } In the proof of Lemma 8.1 we have seen that $\rho _0(j)\rho _0(k)\Phi _{j,k}(x)$ is 2 standard, so the double sum in (8.42) is $${\cal O}(1){1\over\inf \rho _u\rho _0}{1\over\inf \rho _v\rho _0}\vert \rho _u(j)\partial _{x_j}u\vert _2\vert \rho _v(k)\partial _{x_k}v\vert _2.$$ Combining this with (8.37) and the analogous estimate for $v$, we get \ekv{8.43} { {\rm I}_1={\cal O}(1)h{1\over \inf \rho _u\rho _0}{1\over \inf \rho _v\rho _0}. } \smallskip \par\noindent \it Estimate of $I_{2,1}$ (see (8.21)). \rm We write the integrand in (8.21) as \ekv{8.44} { \sum_{j,k,\nu }(\rho _0(j)\rho _0(\nu )\partial _{x_\nu }\Phi _{j,k}){1\over\rho _u(j)\rho _0(j)}(e^{-\phi /2h}\rho _u(j)\partial _{x_j}\partial _{x_k}u){1\over \rho_v(\nu )\rho _0(\nu ) }(\rho _v(\nu )\widetilde{v}_\nu ). } Here we observe that $\rho _0(j)\rho _0(\nu )\partial _{x_\nu }\Phi _{j,k}$ is (3,$\infty $) standard. If $a$ is (2,2) standard, then $\vert a\vert _2={\cal O}(1)$ by Lemma B.1. By Lemma B.2 in the same section, we know that if $b$ is (3,$\infty$) standard, $a$ a 2-tensor and $c$ a 1 tensor, then $$\langle b,a\otimes c\rangle ={\cal O}(1)\vert a\vert _2\vert c\vert _2.$$ Hence the expression (8.44) is \ekv{8.45} { {\cal O}(1){1\over \inf (\rho _u\rho _0)}e^{-\phi /2h}{\cal O}(1){1\over\inf (\rho _v\rho _0) }\vert \rho _v(\nu )\widetilde{v}_\nu (x)\vert _2. } We use this and (8.40) in (8.21), and get \ekv{8.46} { {\rm I}_{2,1}={\cal O}(1){h^2\over\inf (\rho _u\rho _0)\inf (\rho _v\rho _0)}. } \smallskip \par\noindent \it Estimate of $I_{2,2}$ (see (8.21)). \rm Rewrite the integrand in (8.21) as \ekv{8.47} {\sum_{j,k,\nu }\rho _0(j)\Phi _{j,k}e^{-\phi /2h}{1\over \rho _0(j)\rho _u(j)}(\rho _u(j)\partial _{x_\nu }\partial _{x_j}\partial _{x_k}u)\widetilde{v}_\nu =e^{-\phi /2h}\sum_{j,k}\rho _0(j)\Phi _{j,k}B_{j,k},} with \ekv{8.48} {B_{j,k}={1\over \rho _0(j)\rho _u(j)}\sum_{\nu }(\rho _u(j)\partial _{x_\nu }\partial _{x_j}\partial _{x_k}u)\widetilde{v}_\nu .} Here we recall that $\rho _u(j)\partial _{x_\nu }\partial _{x_j}\partial _{x_k}u$ is (3,2) standard, so if we view $B=(B_{j,k})$ as a matrix, \ekv{8.49} { \Vert B\Vert _{{\cal L}(\ell^\infty ,\ell^1)}={\cal O}(1){1\over\inf (\rho _0\rho _u)}\vert \widetilde{v}_\nu \vert _2. } On the other hand $\rho _0(j)\Phi _{j,k}$ is 2 standard and hence ${\cal O}(1):\ell^1\to \ell^1$. If we view the last sum in (8.47) as ${\rm tr\,}((\rho _0(j)\Phi _{j,k})\circ {^t\hskip -1pt B})$, we conclude by the trace lemma that it is ${\cal O}(1){1\over\inf (\rho _0\rho _u)}\vert \widetilde{v}_\nu \vert _2$. Using this in (8.47) and then in (8.21), we get \ekv{8.50} { {\rm I}_{2,2}={\cal O}(h^{3\over 2}){1\over \inf (\rho _0\rho _u)}\Vert \widetilde{v}\Vert _{\ell^2\otimes L^2}={\cal O}(h^2){1\over\inf (\rho _0\rho _u)}. } Here we used (8.40) in the last step, with $\rho _v$ replaced by 1. By playing with $\rho _v$ also we could reach the estimate $${\rm I}_{2,2}={\cal O}(h^2){1\over\inf (\rho _0\rho _u)}{1\over\inf (\rho _0\rho _v)},$$ provided that we add to (H6), the assumption that $$\sum_{j,k,\nu }\rho _u(j){\rho _v(k)\over\rho _v(\nu )}(\partial _{x_\nu }\partial _{x_j}\partial _{x_k}u) t_js_kr_\nu ={\cal O}(1)\vert t\vert _\infty \vert s\vert _\infty \vert r\vert _2.$$ \smallskip \par\noindent \it Estimate of $II_{1,1}$ (see (8.25)). \rm Write the integrand in (8.25) as $$\sum_{j,\nu }{1\over\rho _0(j)\rho _u(j)}(\rho _0(j)\partial _{x_\nu }\Psi _j)(e^{-\phi /2h}\rho _u(j)\partial _{x_j}u)\widetilde{v}_\nu ={{\cal O}(1)\over\inf (\rho _0\rho _u)}e^{-\phi /2h}\vert \widetilde{v}_\nu \vert _2.$$ Here we used that $\rho _0(j)\partial _{x_\nu }\Psi _j$ is 2 standard by Lemma 8.1, and that $\rho _u(j)\partial _{x_j}u$ is (1,2) standard by (H6). Inserting this into (8.25) and using (8.40) with $\rho _v$ replaced by 1, we get \ekv{8.51} { {\rm II}_{1,1}={\cal O}(1){h^2\over\inf (\rho _0\rho _u)}. } \smallskip \par\noindent \it Estimate of $II_{1,2}$ (see (8.25)). \rm Write the integrand in (8.25) as \ekv{8.52} { e^{-\phi /2h}\sum_{j,\nu }(\rho _0(j)\Psi _j){1\over \rho _0(j)\rho _u(j)}(\rho _u(j)\partial _{x_\nu} \partial _{x_j}u)\widetilde{v}_\nu ={\cal O}(1)e^{-\phi /2h}{1\over\inf (\rho _0\rho _u)}\vert \widetilde{v}_\nu \vert _2, } where we used Lemma 8.1 and (H6). Using this with (8.40) in (8.25), we get \ekv{8.53} { {\rm II}_{1,2}={\cal O}(1){h^2\over\inf (\rho _0\rho _u)}. } \smallskip \par\noindent \it Estimate of $III_{1,1}$ (see (8.29)). \rm We write the integrand in (8.29) as \ekv{8.54} { \sum_{\nu ,\mu }\widetilde{v}_\mu (\rho _1(\nu )Z_\mu \widetilde{D}_\nu ){1\over \rho _1(\nu )\rho _u(\nu )}(\rho _u(\nu )\partial _{x_\nu }u)={{\cal O}(1)\over\inf (\rho _1\rho _u)}\vert \widetilde{v}\vert _2\vert \rho _1(\nu )Z_\mu \widetilde{D}_\nu \vert _{\ell^2\otimes \ell^2}, } where we used (H6). Using this in the expression for ${\rm III}_{1,1}$ together with (8.40), (8.41), we get \ekv{8.55} { {\rm III}_{1,1}={\cal O}(1){h^{3\over 2}\over\inf (\rho _1\rho _u)}. } \smallskip \par\noindent \it Estimate of $III_{1,2}$ (see (8.29)). \rm Write the integrand as \ekv{8.56} { \sum_{\nu ,\mu }\widetilde{v}_\mu (\rho _1(\nu )\widetilde{D}_\nu ){1\over \rho _1(\nu )\rho _u(\nu )}(\rho _u(\nu )\partial _{x_\mu }\partial _{x_\nu }u)={\cal O}(1){1\over\inf (\rho _1\rho _u)}\vert \widetilde{v}_\mu \vert _2\vert \rho _1(\nu )\widetilde{D}_\nu \vert _2, } since $\vert \rho _u(\nu )\partial _{x_\mu }\partial _{x_\nu }u\vert _{\ell^2\otimes \ell^2}={\cal O}(1)$ by (H6) and Lemma B.1. Using this in (8.29) with (8.40), (8.41), we get \ekv{8.57} {{\rm III}_{1,2}={\cal O}(1){h^2\over\inf (\rho _1\rho _u)}.} \par Recall that for $X={\rm I},{\rm II},{\rm III}$ and $i=1,2$, we get $X_{2,i}$ from $X_{1,i}$ by exchanging $u$ and $v$ as well as their associated quantities. This means that we get the estimates for $X_{2,i}$ from those for $X_{1,i}$, by exchanging $u$ and $v$ to the right, and we therefore obtain estimates for all terms in (8.30). Summing up, we have \medskip \par\noindent \bf Proposition 8.4. \it Let $\phi _t(x)=\phi _t(x;h)$, $0\le t\le 1$, $x\in{\bf R}^\Lambda $ be $C^1$ in $t$ and smooth in $x$, of the form (8.1H), satisfying (8.2H). Let $W$ be a set of weights $\rho :\Lambda \to ]0,\infty [$, with $\rho \in W\Rightarrow 1/\rho \in W$, $1\in W$. Assume that $\phi =\phi _t$ satisfies ($\widetilde{H1}$) (section 7), (H2), (H3) (section 1), ($\widetilde{H4}$) and (H5) of this section uniformly with respect to $t\in [0,1]$. Here $W_a$ is defined prior to (H5) with some fixed $a$ as in (8.32). Let $u,v\in C^\infty ({\bf R}^\Lambda ;{\bf R})$ be independent of $t$ and satisfy (H6). Finally choose $\rho _0\ge \rho _1\in W_a$ such that (H7) holds. (Cf Lemma 8.1.) Then \ekv{8.58} {\partial _t{\rm Cor}_{\phi _t}(u,v)={\cal O}(1)({h\over \inf (\rho _u\rho _0)\inf (\rho _v\rho _0)}+{h^{3\over 2}\over\inf (\rho _1\rho _u)}+{h^{3\over 2}\over \inf (\rho _1\rho _v)}).}\rm\medskip \par The estimate (8.58) could certainly be improved to the price of some further assumptions. Also notice that the assumptions of standardness could be weakened, since we only use derivatives up to some fixed finite order. %\vfill\eject \bigskip \centerline{\bf 9. Asymptotics of the correlations.} \medskip \par This section is divided into three parts. In part A we make only the assumptions of section 1 and consider the correlation of two functions with (1,2)-standard gradients, which are independent of $h$. We show that it has an asymptotic expansion in powers of $h$, and that this expansion is valid uniformly with respect to $\Lambda $. In part B we let $\Lambda =({\bf Z}/L{\bf Z})^d$ with $L\in \{ 2,3,..\}$. Adding assumptions on $\phi ''(0)$; an assumption of translation invariance, as well as the assumption ($\widetilde{{\rm H}1}$) of section 7 for a suitable family of weights, we study the asymptotics of ${\rm Cor\,}(x_\nu ,x_\mu )$ for $\nu ,\mu \in \Lambda $, when $1\ll {\rm dist\,}(\nu ,\mu )\ll L$ and obtain the product of an exponentially decaying factor and a factor with power behaviour, in the limit $\nu -\mu \to \infty $. The exponent in the exponential factor is positively homogeneous of degree $1$ in $\nu -\mu $ and we show that it has an asymptotic expansion in powers of $h$. We obtain a similar result for the power factor. In this result all terms in the asymptotic expansions, may depend on $\Lambda $ but they remain bounded and the asymptotic expansions are valid uniformly in $\Lambda $. In part C we make some additional assumptions that allow us to pass to the thermodynamical limit. This part contains the final result of the paper, and the results here remain valid also with $\Lambda $ equal to a finite subset of ${\bf Z}^d$ which contains a large ball centered at $0$. (In section 10 we derive simplified sets of assumptions in order to reach the formulation of the main result in the introduction.) Throughout the whole section we make the assumptions (H1--4) of section 1 and let the functions $\phi $ be normalized as in (8.2H). \smallskip \par\noindent $\underline{{\rm A.}}$ In this part we only assume that $u$, $v$ are functions on ${\bf R}^\Lambda $ independent of $h$, such that $\nabla u$, $\nabla v$ are (1,2) standard (as defined in section 8). We are interested in the asymptotics of \ekv{9.1} { {\rm Cor\,}(u,v)=(e^{-\phi /2h}(u-\langle u\rangle )\vert e^{-\phi /2h}(v-\langle v\rangle ))=h({\Delta _\phi ^{(1)}}^{-1}e^{-\phi /2h}du\vert e^{-\phi /2h}dv), } as $h\to 0$. The second equality was established in this explicit form in [S1], but already effectively used in earlier work by Helffer and the author [HS]. Under the present assumptions the derivation is very simple: Let $\widetilde{u}\in{\cal S}$ solve (8.18). After an integration by parts, we get ${\rm Cor\,}(u,v)=(d_\phi \widetilde{u}\vert d_\phi (e^{-\phi /2h}v))$. In view of (8.38), we have $d_\phi \widetilde{u}=(\Delta _\phi ^{(1)})^{-1}d_\phi u$, which gives the last expression in (9.1). \par We apply Proposition A.1, which extends to ($P$,2) standard tensors, and write \ekv{9.2} { du\,e^{-\phi /2h}=\sum_{n=0}^{N+1}\sum_{\nu =0}^M\sum_{j\in\Lambda ^n}h^{{1\over 2}n+\nu }Z^*_j(\widetilde{u}^n_{j,\nu }e^{-\phi /2h}), } where $\widetilde{u}^n_{j,\nu }$ is ($1+j$,2) standard, and for $\nu 0\hbox{ for } j\in K,\hbox{ and }{\rm Gr}(K)={\bf Z}^d, } where ${\rm Gr}(K)$ denotes the smallest subgroup of ${\bf Z}^d$ which contains $K$. Put \ekv{9.14} { F_{\widetilde{v}_0}(\eta )=\sum_{k\in {\bf Z}^d}e^{k\cdot \eta }\widetilde{v}_0(k),\ \eta \in{\bf R}^d, } where we know ([S1]) that \ekv{9.15} {\{ \eta \in {\bf R}^d;\, F_{\widetilde{v}_0}(\eta )<\infty \}} is convex and that $F_{\widetilde{v}_0}$ is a convex function, which is smooth on the interior of the set (9.15). Assume \eekv{{\rm H}11} { \exists \hbox{ an open convex even set }\widetilde{\Omega }\subset{\bf R}^d,\hbox{ independent of }\Lambda ,\hbox{ such that }} {F_{\widetilde{v}_0}(\eta ) \hbox{ is uniformly bounded on every compact subset of }\widetilde{\Omega }. } Here we define even sets to be the ones which are symmetric around 0. Then from [S1] (based on the fact that $K$ is contained in no hyperplane of ${\bf R}^d$) we know that $F_{\widetilde{v}_0}$ is strictly convex: \ekv{9.16} { \nabla ^2F_{\widetilde{v}_0}(\eta )\ge {1\over{\cal O}(1)},\ \eta \in\widetilde{\Omega }. } (We even have that $\log F_{\widetilde{v}_0}$ is strictly convex.) \par Let $\Omega \subset\subset\widetilde{\Omega }$ be an open even convex set, which is independent of $\Lambda $ and assume that \ekv{9.17{\rm H}} {\liminf_{\Omega \ni \eta \to\partial \Omega }F_{\widetilde{v}_0}(\eta )\ge 1+3\epsilon _0,} where $\epsilon _0>0$ is independent of $\Lambda $. $F_{\widetilde{v}_0}(\eta )$ is then uniformly bounded in $\Omega $ and its derivatives are uniformly bounded on every fixed relatively compact subset of $\Omega $. Using also (9.16), it is clear that the sets \ekv{9.18} { \Omega _b:=\{ \eta \in\Omega ; F_{\widetilde{v}_0}0$ arbitrarily large, provided that we choose $\epsilon _1$ small enough. Combining this with (9.20), we see as in [BJS] (and [S1], [SW]) \ekv{9.21} {\sum_{x\in{\bf Z}^d}e^{\widetilde{r}(x)}\widetilde{v}_0(x)\le 1+\epsilon _0.} \par Let $r\in C^{1,1}(({\bf R}/L{\bf Z})^d)$ be real and assume that $\nabla r(x)\in\Omega _{1+\epsilon _0/2}$, $\vert \nabla ^2r\vert \le \epsilon _1$ everywhere, where $\epsilon _1$ is small enough. Then \ekv{9.22} { \sum_{x\in\Lambda }e^{r(x)}v_0(x)=\sum_{x\in{\bf Z}^d}e^{r\circ \pi _\Lambda (x)}\widetilde{v}_0(x)\le 1+\epsilon _0, } since $\widetilde{r}:=r\circ \pi _\Lambda $ satisfies the earlier assumptions. \par If $r'(x)$ is merely Lipschitz on $({\bf R}/L{\bf Z})^d$ with $r'(0)=0$, and $\nabla r'(x)\in\Omega _{1+\epsilon _0/2}$ a.e., then by regularization,we can find $r$ with the above properties, such that $r-r'={\cal O}_{\epsilon _0,\epsilon _1}(1)$. \par Let $a=-\epsilon _0$, and let $W=W_a$ consist of all weights $\rho (x)=e^{r(x)}$, $x\in ({\bf R}/L{\bf Z})^d$, for which $\nabla r\in\Omega _{1+\epsilon _0/2}$, and $\vert \nabla ^2r\vert \le \epsilon _1$, with $\epsilon _1>0$ sufficiently small. Using Shur's lemma and (9.22) (with $r$ there replaced by $r(x)-r(0)$) we see that for all $\rho \in W$: \ekv{9.23} { \Vert \rho (v_0*)\rho ^{-1}\Vert _{{\cal L}(\ell^2,\ell^2)}, \Vert \rho^{-1} (v_0*) \rho \Vert _{{\cal L}(\ell^2,\ell^2)}\,\le 1+\epsilon _0,} so that \ekv{9.24} { (\rho ^{-1}\phi ''(0)\rho u\vert u)\ge -\epsilon _0\vert u\vert ^2,\ u\in\ell^2(\Lambda ). } We fix $\epsilon _0$ with \ekv{9.25} { 0<\epsilon _0<1-\vert v_0\vert _1-{1\over{\cal O}(1)}, } so that \ekv{9.26} {-\epsilon _0>-\lambda _{\min}(\phi ''(0))+{1\over{\cal O}(1)}=-1+\vert v_0\vert _1+{1\over{\cal O}(1)}.} \par As for the higher derivates of $\phi $, we will assume ($\widetilde{{\rm H}1})$ (section 7), with $W$ equal to the set of weights defined above. \par We apply Proposition 7.1 with $N=0$, $z=0$, and we shall drop the superscript (0,1) for simplicity. We recall the formula \ekv{9.27} { {\Delta _\phi ^{(1)}}^{-1}=E-E_+E_{-+}^{-1}E_-, } that we shall use in (9.1), with $u=x_\mu $, $v=x_\nu $, $\mu ,\nu \in\Lambda $. \par Let us first consider the contribution from $E$. According to Proposition 7.1, we know that $(\rho ^{-1}\otimes 1)E(\rho \otimes 1)={\cal O}(1):\ell^2\otimes L^2\to \ell^2\otimes L^2$, for all $\rho \in W$, and in view of the observation after (9.22), we know that this remains true for $\rho =e^r$, with $r\in{\rm Lip\,}(({\bf R}/L{\bf Z})^d)$, $\nabla r\in\Omega _{1+\epsilon _0/2}$ a.e. Now recall that we have introduced the norm $p_b$ on ${\bf R}^d$ in (9.19). Let $d_b=d_b^\Lambda $ be the corresponding distance on $\Lambda $, given by $$d_b(\nu ,\mu )=\inf_{\widetilde{\nu }\in \pi _\Lambda ^{-1}(\nu )\atop \widetilde{\mu }\in \pi _\Lambda ^{-1}(\mu )}p_b(\widetilde{\nu }-\widetilde{\mu }).$$ Then $(\rho _\mu \otimes 1)e^{-\phi /2h}dx_\mu ={\cal O}(1)$ in $\ell^2\otimes L^2$ with $\rho _\mu =e^{d_{1+\epsilon _0/2}(\mu ,\cdot )}$. By the weighted boundedness result for $E$, that we have established above, we have $(\rho _\mu \otimes 1)E(e^{-\phi /2h}dx_\mu )={\cal O}(1)$ in $\ell^2\otimes L^2$. It follows that (cf. (9.1)) \ekv{9.28} { h(Ee^{-\phi /2h}dx_\mu \vert e^{-\phi /2h}dx_\nu )={\cal O}(1)he^{-d_{1+\epsilon _0/2}(\nu ,\mu )}. } As a matter of fact, since $e^{-\phi /2h}dx_\mu $, $e^{-\phi /2h}dx_\nu $ belong to the image of $R_-$ and $ER_-=0$, the expression (9.28) vanishes, However the weaker formulation in (9.28) may be of interest for more general correlations. The main contribution to (9.1) will come from the second term in (9.27) and is equal to \ekv{9.29} { -h(E_{-+}^{-1}E_-(e^{-\phi /2h}dx_\mu )\vert E_-(e^{-\phi /2h}dx_\nu ))=-h(E_{-+}^{-1}\delta _\mu ,\delta _\nu ), } since $e^{-\phi /2h}dx_\mu =R_-\delta _\mu $ and $E_-R_-=1$. Here we are in the presence of convolution matrices. Indeed, from (H8) we deduce (cf. [S1]) a certain translation invariance for ${\cal P}^{0,1}$ and its inverse, which implies that $E_{-+}$ and its inverse are convolutions and \ekv{9.30} { E_-(e^{-\phi /2h}dx_\mu )=\tau _\mu E_-(e^{-\phi /2h}dx_0). } Proposition 7.4 can be applied together with the remark after (9.22) to show that \ekv{9.31} { E_{-+}= -(1-v*),\ v\sim\sum_{\nu =0}^{\infty }h^\nu v_\nu \hbox{ in }{\cal L}(\rho \ell^2,\rho \ell^2), } uniformly when $\rho =e^r$, with $r$ Lipschitz, such that $\nabla r\in\Omega _{1+\epsilon _0/2}$ a.e., with $v_0$ as before. Here we equip $\rho \ell^2$ with the natural norm $\Vert \rho u\Vert _{\rho \ell^2}=\Vert u\Vert _{\ell^2}$. This implies that \ekv{9.32} { \vert v(\ell )\vert \le {\cal O}(1)e^{-d_{1+\epsilon _0/2}(\ell )},\ \vert v_\nu (\ell )\vert \le {\cal O}_\nu (1)e^{-d_{1+\epsilon _0/2}(\ell )},\ \ell\in\Lambda . } \par We have already assumed that $v_0$ has a lift $\widetilde{v}_0$ to ${\bf Z}^d$ with certain properties including (9.13) and we know that $\widetilde{v}_0(\ell )={\cal O}(1)e^{-p_{1+\epsilon _0/2}(\ell)}$. For $v_{\nu}$, $\nu \ge 1$ and $v-v_0$ we use the following lift: If $\ell\in\Lambda $, let $A(\ell )\subset{\bf Z}^d$ be the set of $\widetilde{\ell }$ in $\pi _\Lambda ^{-1}(\ell )$ for which $p_{1+\epsilon _0/2}(\widetilde{\ell})$ is minimal ($=d_{1+\epsilon _0/2}(0,\ell )$). Let $\widetilde{\Lambda }$ be the union of all such $A(\ell )$, and define $\widetilde{v}_\nu (\ell )$ to be the unique function on ${\bf Z}^d$ with support in $\widetilde{\Lambda }$, such that \ekv{9.33} { v_\nu (\ell )=\sum_{\widetilde{\ell}\in\pi _\Lambda ^{-1}(\ell )}\widetilde{v}_\nu (\widetilde{\ell}), } and such that $\widetilde{v}_\nu $ is constant on each $A(\ell )$. Define $\widetilde{v}-\widetilde{v}_0$ (where $\widetilde{v}_0$ is already known) by the same construction. This means that we have defined $\widetilde{v}$. Then \ekv{9.34} { v(\ell )=\sum_{\widetilde{\ell}\in\pi _\Lambda ^{-1}(\ell )}\widetilde{v}(\ell ), } \ekv{9.35} { \widetilde{v}\sim \sum_0^\infty h^\nu \widetilde{v}_\nu } in $\ell^\infty $ and even in $e^{-p_{1+\epsilon _0/2}}\ell ^\infty $. Let \ekv{9.36} { \widehat{\widetilde{v}}(\xi )=\sum_{\ell\in{\bf Z}^d}\widetilde{v}(\ell )e^{-i\ell \xi },\ \xi \in ({\bf R}/2\pi {\bf Z})^d=:{\bf T}^d, } denote the Fourier transform of $\widetilde{v}$. From the above asymptotic expansions, it follows that $\widehat{\widetilde{v}}$ extends to a holomorphic function in ${\bf T}^d+i\Omega _{1+\epsilon _0/2}$ which is uniformly bounded and has a uniform asymptotic expansion \ekv{9.37} { \widehat{\widetilde{v}}(\zeta )\sim\sum_{\nu =0}^\infty h^\nu \widehat{\widetilde{v}}_\nu (\zeta ), } in ${\bf T}^d+i\Omega _b$, for every fixed $b<1+\epsilon _0/2$. \par As in [S1], we can study the asymptotic behaviour of $(1-\widetilde{v}*)^{-1}$ by means of Fourier inversion. In that paper (as well as in [BJS]) we only knew that $\widehat{\widetilde{v}}(\zeta )=\widehat{\widetilde{v}}_0(\zeta )+{\cal O}(h^{1/2})$ in ${\bf T}^d+i\Omega _b$, while we now have the full asymptotic expansion (9.37), but the discussion in the above mentioned papers goes through without any essential changes and will give the full $h$ asymptotics. We only recall some steps. If \ekv{9.38} { \widetilde{F}*=(1-\widetilde{v}*)^{-1}, } then \ekv{9.39} { \widetilde{F}(k)={1\over (2\pi )^d}\int_{{\bf T}^d}{e^{ik\cdot \xi }\over 1-\widehat{\widetilde{v}}(\xi )}d\xi . } In section 4 of [S1], we discussed the corresponding inverse $F_0*$ of $(1-\widetilde{v}_0*)$, \ekv{9.40} {\widetilde{F}_0(k)={1\over (2\pi )^d}\int_{{\bf T}^d}{e^{ik\cdot \xi }\over 1-\widehat{\widetilde{v}}_0}d\xi ,} denoted by $E(k)$ there. We observed that $1-\widehat{\widetilde{v}}_0(\zeta )\ne 0$ for $\zeta =\xi +i\eta $, with $\eta \in \Omega _1$, and that $1-\widehat{\widetilde{v}}_0(\zeta )$ vanishes for $\eta \in\partial \Omega _1$ precisely for $\xi =0$. (Here is where the full power of (H10) is used.) We do not repeat the proof here, but simply recall that $\widehat{\widetilde{v}}_0(i\eta )=F_{\widetilde{v}_0}(\eta )$. For $k\in{\bf R}^d\setminus\{ 0\}$, let $\eta _0(k)=\eta _0(k/\vert k\vert )\in\partial \Omega _1$ be the unique point where the exterior normal of $\Omega _1$ is equal to a positive multiple of $k$. We can write \ekv{9.41} { \eta _0(k)=\eta _0'(k)+p_1({k\over\vert k\vert }){k\over \vert k\vert},\ \eta _0'(k)\in (k)^\perp } and because of the strict convexity, we can represent the boundary of $\partial \Omega _1$ in a neighborhood of $\eta _0(k)$ as \ekv{9.42} { \partial \Omega _1=\{ \eta _0'(k)+\eta '+(p_1({k\over \vert k\vert })-g_{{k\over \vert k\vert },0}(\eta ')){k\over \vert k\vert };\, \eta '\in (k)^\perp\cap{\rm neigh\,}(0)\}, } where $g$ is a real and analytic function vanishing to second order at $0$ and with \ekv{9.43} { g''_{{k\over \vert k\vert },0}(0)>0. } Near $i\eta (k)$, we can view the complex hypersurface $1=\widehat{\widetilde{v}}_0(\zeta )$ as the complexification of the real-analytic hypersurface $i\partial \Omega _1$, so we get \ekv{9.43} { (1-\widehat{\widetilde{v}}_0)^{-1}(0)=\{ i\eta '_0(k)+\zeta '+i(p_1({k\over \vert k\vert })-g_{{k\over \vert k\vert },0}({\zeta '\over i})){k\over \vert k\vert };\, \zeta '\in{\rm neigh\,}(0),\, k\cdot \zeta '=0\} . } By contour deformation and residues, we got in [S1]: \eeekv{9.44} { \widetilde{F}_0(k)=}{{ie^{-p_1(k)}\over (2\pi )^{d-1}}\int_{\xi '\in V\cap (k)^\perp}{e^{\vert k\vert g_{k/\vert k\vert ,0}(-i\xi ')} \over -({k\over \vert k\vert }\cdot \partial )(\widehat{\widetilde{v}}_0)(\xi '+i\eta _0'(k)+i(p_1({k\over \vert k\vert })-g_{{k\over \vert k\vert },0}({\xi\over i }')){k\over \vert k\vert }) }d\xi '}{\hskip 8cm +{\cal O}(1)e^{-p_1(k)-\delta _0\vert k\vert }, } where $V$ is a small real neighborhood of $0$ in ${\bf R}^d$ and $\delta _0>0$ some fixed constant. \par When passing from $\widetilde{v}_0$ to $\widetilde{v}$ very little changes. Let $\Omega _1(h)$ be the set of $\eta \in\Omega _{1+\epsilon _0/2}$ such that $F_{\widetilde{v}}(\eta )\le 1$. This set is very close to $\Omega _1$, and is strictly convex. Again, we have $\widehat{\widetilde{v}}(i\eta )=F_{\widetilde{v}}(\eta )$. For $k\in {\bf R}^d\setminus\{ 0\}$, let $\eta (k)\sim\sum \eta _\nu ({k\over \vert k\vert })h^\nu $ be the unique point in $\partial \Omega _1(h)$, where the exterior unit normal of $\Omega _1(h)$ is equal to a positive multiple of $k$. Write \ekv{9.45} { \eta (k)=\eta '(k)+p_{1,h}({k\over \vert k\vert }){k\over \vert k\vert },\ \eta '(k)\in (k)^\perp, } where $p_{1,h}(k)=\sup_{\eta \in\Omega _1(h)}k\cdot \eta $ is the support function of $\Omega _1(h)$. Then \ekv{9.46} { p_{1,h}\sim p_1+\sum_{\nu =1}^\infty p_1^{(\nu )}(k), } with $p_1^{(\nu )}$ positively homogeneous of degree 1. In a neighborhood of $\eta (k)$, we can represent $\partial \Omega _1(h)$ as \ekv{9.47} { \partial \Omega _1(h)=\{ \eta '(k)+\eta '+(p_{1,h}({k\over \vert k\vert })-g_{{k\over \vert k\vert }}(\eta ')){k\over \vert k\vert };\, \eta '\in (k)^\perp \cap{\rm neigh\,}(0)\}, } where $g$ is real and analytic and has the uniform asymptotic expansion \ekv{9.48} { g_{k\over \vert k\vert }(\eta ')\sim\sum_{\nu =0}^\infty g_{{k\over \vert k\vert },\nu }(\eta ')h^\nu , } for $\eta '\in (k)^\perp\cap\widetilde{V}$, where $\widetilde{V}$ is a fixed complex neighborhood of $0$. \par Again, as in [S1], we get \eeekv{9.49} { \widetilde{F}(k)=}{{ie^{-p_{1,h}(k)}\over (2\pi )^{d-1}}\int_{\xi '\in V\cap (k)^\perp}{e^{\vert k\vert g_{k/\vert k\vert }({\xi '\over i};h)} \over -({k\over \vert k\vert }\cdot \partial )(\widehat{\widetilde{v}})(\xi '+i\eta '(k)+i(p_{1,h}({k\over \vert k\vert })-g_{{k\over \vert k\vert }}({\xi '\over i})){k\over \vert k\vert }) }d\xi '}{\hskip 8cm +{\cal O}(1)e^{-p_{1,h}(k)-\delta _0\vert k\vert }, } with $V$ and $\delta _0$ as in (9.44). $g_{k\over \vert k\vert }({\xi '\over i};h)$ vanishes to second order at $\xi '=0$ and\break ${\rm Re\,}{\rm Hess\,}g_{{k\over \vert k\vert }}(0;h)\le -{\rm Const.}<0$. The method stationary phase gives the large $k$ asymptotics of $F(k)$ uniformly in $\Lambda $ and in $h$ (for $h\le h_0>0$ sufficiently small), where all the involved functions have uniform asymptotic expansions in powers of $h$: \ekv{9.50} { \widetilde{F}(k)={\cal O}(1) e^{-(p_{1,h}(k)+\delta _0\vert k\vert )}+e^{-p_{1,h}(k)}q(k;h), } \ekv{9.51} { q(k;h)\sim\sum_{-\infty }^0 q_{-{d-1\over 2}-\nu }(k;h),\ k\to\infty , } where $q_j(k;h)$ is smooth and positively homogeneous of degree $j$ in $k$ and has an asymptotic expansion, \ekv{9.52} {q_j(k;h)\sim\sum_{\nu =0}^\infty h^\nu q_{j,\nu }(k),\ h\to 0 ,} where $q_{j,\nu }$ is also positively homogeneous of degree $j$. In [S1], the leading term was computed: \ekv{9.53} { q_{-{d-1\over 2},0}(k)={1\over (2\pi \vert k\vert )^{{d-1\over 2}}}\,{(({k\over \vert k\vert }\cdot \partial _\eta )F_{\widetilde{v}_0}(\eta (k)))^{{d-3\over 2}} \over\sqrt{ \det (\partial _\eta '^2F_{\widetilde{v}_0})(\eta (k)) } }, } where $\eta '$ indicates some orthonormal coordinates in $(k)^{\perp}$. \par The convolution operator $1-v*$ on $\Lambda $ has the inverse $F*$, where \ekv{9.54} { F(k)=\sum_{\widetilde{k}\in\pi _\Lambda ^{-1}(k)}\widetilde{F}(\widetilde{k}). } For $\delta >0$, let $\Lambda _\delta $, be the set of $k\in\Lambda $, such that \smallskip \par\noindent $\rm 1^o$ there is a unique $\widetilde{k}(k)\in\pi _\Lambda ^{-1}(k)$, such that $d_1(k,0)=p_1(\widetilde{k}(k))$, \smallskip \par\noindent $\rm 2^o$ $p_1(\ell )\ge (1+\delta )d_1(k,0)$, whenever $\widetilde{k}(k)\ne \ell\in\pi _\Lambda ^{-1}(k)$.\smallskip Fix $\delta >0$. Then from (9.50), (9.54) we get the uniform asymptotics for $k\in\Lambda _\delta $: \ekv{9.55} { F(k)={\cal O}(1)e^{-(d_{1,h}(0,k)+\delta _0\vert \widetilde{k}(k)\vert )}+e^{-d_{1,h}(0,k)}q(\widetilde{k}(k);h), } where $d_{1,h}$ is the distance on $\Lambda $, induced by $p_{1,h}$ and $\delta _0>0$ a constant. We also have the bound \ekv{9.56} { F(k)={\cal O}(1) \vert k\vert ^{-{d-1\over 2}}e^{-d_{1,h}(0,k)},\ k\in\Lambda . } Since $E_{-+}^{-1}=-F*$, (9.55), (9.56) give us an asymptotic expansion for $E_{-+}^{-1}(\mu ,\nu )$ for $\mu -\nu \in\Lambda _\delta $ and a precise upper bound for all $\mu ,\nu \in\Lambda $. \par Before using this, it may be instructive to study (cf. (9.30)) \ekv{9.57} {E_-(e^{-\phi /2h}dx_0),} having in mind also more general correlations. Since $E_-=E_+^*$, we can apply Proposition 7.4 and conclude that \eekv{9.58} { E_-(e^{-\phi /2h}dx_0)\equiv (C^M)^*R_+^{M,1}(e^{-\phi /2h}dx_0)\equiv (C^M)^*(\delta _0)} {\hskip 4cm \equiv\sum_{\nu =0}^{\widetilde{M}}h^\nu D_{0,0}^{M,\nu }(\delta _0\otimes e_0)+{\cal O}(1)h^{M/2},} modulo ${\cal O}(h^{M/2})$ in $e^{-d_{1+\epsilon _0/2}}\ell^2$, where $e_0\in \ell_b^2(\Lambda ^0\cup\Lambda ^1\cup ..\cup\Lambda ^N)$ is the element given by $1\in{\bf C}\simeq \ell^2(\Lambda ^0)$. Here $M$ can be chosen arbitrarily large, so we get a full asymptotic expansion \ekv{9.59} { E_-(e^{-\phi /2h}dx_0)\sim\sum_{\nu =0}^\infty h^\nu f_\nu \hbox{ in }e^{-d_{1+\epsilon _0/2}}\ell^2, } where we also know that $f_0=\delta _0$. \par Now we combine this with (9.31), (9.27), (9.28), (9.29), (9.55), (9.56) and get \medskip \par\noindent \bf Proposition 9.2. \it Assume (H1--4) of section 1, (H8--11) of this section, (9.17H), and ($\widetilde{{\rm H}1}$) of section 7 for the set of weights $\rho =e^{r(x)}$ with $\nabla r(x)\in\Omega _{1+\epsilon _0/2}$ a.e. (discussed after (9.22)). Then \ekv{9.60} {{\rm Cor\,}(x_\nu ,x_\mu )={\cal O}(h){\rm dist\,}(\nu ,\mu )^{-{d-1\over 2}}e^{-d_{1,h}(\nu ,\mu )},\ \nu ,\mu \in\Lambda ,} \ekv{9.61} {{\rm Cor\,}(x_\nu ,x_\mu )={\cal O}(h) e^{-(d_{1,h}(\nu ,\mu )+\delta _0{\rm dist\,}(\nu ,\mu ))}+he^{-d_{1,h}(\nu ,\mu )}q(\widetilde{k}(\nu -\mu );h),\ \nu ,\mu \in\Lambda _\delta , } where $q$, $\widetilde{k}$, $d_{1,h}$ have been defined above (cf. (9.50), (9.55)).\rm \medskip \par\noindent $\underline{\rm C.}$ Let $U_j$, $V_j$ be increasing sequences of bounded subsets of ${\bf Z}^d$ with $U_j\subset V_j$, $U_j\nearrow {\bf Z}^d$, $j\to \infty $. Let $\rho _0=\rho _{0,j}:{\bf Z}^d\to ]0,\infty [$ be the weight \ekv{9.62} { \rho _{0,j}(\nu )=\exp (\theta \,{\rm dist\,}(\nu ,{\bf Z}^d\setminus U_j)), } for some fixed (possibly small) $\theta >0$, and where ${\rm dist}$ denotes the standard Euclidean distance on ${\bf Z}^d$. Let $\phi =\phi _j\in C^\infty ({\bf R}^{V_j};{\bf R})$ satisfy the assumptions of section 1 (with $V_j=\Lambda $) and assume \eekv{{\rm H}12} { \hbox{If }k>j,\hbox{ then }(\rho _0\otimes \rho _0)(\phi _j\oplus\psi _{k,j}-\phi _k)''\hbox{ is 2 standard,} } { \hbox{if }\psi _{k,j}\hbox{ is defined on }{\bf R}^{V_k\setminus V_j}\hbox{ with }\psi _{k,j}''\hbox{ 2 standard.} } Notice that this condition is independent of the choice $\psi _{k,j}$, and we could for instance just take zero. \par We also assume that $\phi =\phi _j$ satisfies ($\widetilde{{\rm H}1}$) (section 7), ($\widetilde{{\rm H}4}$) (section 8) with \ekv{9.63} {W=\{ \rho =e^r;\, \vert r(\nu )-r(\mu )\vert \le \theta \vert \nu -\mu \vert \} ,\ \vert \cdot \vert =\vert \cdot \vert _{\ell^2},} so that $\rho _0\in W$. Let \ekv{9.64} { \rho _1(\nu )=\rho _0(\nu ) e^{-\theta \vert \nu \vert /4},} \ekv{9.65} {S=S_j=\{ \nu \in U_j;\, \vert \nu \vert \le {\rm dist\,}(\nu ,{\bf Z}^d\setminus U_j)\},\ r_j={\rm dist\,}(0,{\bf Z}^d\setminus U_j),} so that \ekv{9.66} { \rho _1(\nu )\ge e^{\theta r_j/4},\ \nu \in S_j. } Put $$\phi _{j,k,t}=t\phi _k+(1-t)(\phi _j\oplus \psi _{k,j}),\ 0\le t\le 1,$$ with $\psi _{k,j}(x)=\sum_{\nu \in V_k\setminus V_j}x_\nu ^2$, where we drop the normalization constant (cf. (8.1H)) for simplicity. Assume that $\phi _{j,k,t}$ satisfies ($\widetilde{{\rm H}4}$) with $W$ given in (9.63). Then we can apply Proposition 8.4 and obtain for $\nu ,\mu \in S_j$: \ekv{9.67} {{\rm Cor\,}_{\phi _k}(x_\nu ,x_\mu )-{\rm Cor\,}_{\phi _j}(x_\nu ,x_\mu )={\cal O}(1)h e^{-\theta r_j/4}.} Here we also used that $${\rm Cor\,}_{\phi _j}(x_\nu ,x_\mu )={\rm Cor\,}_{\phi _j\oplus\psi _{k,j}}(x_\nu ,x_\mu ).$$ \par Let $\Lambda =\Lambda _j=({\bf Z}/L_j{\bf Z})^d$ be a sequence of discrete tori with $L_j\nearrow \infty $ large enough so that there exists a natural embedding \ekv{9.68} { V_j\subset\Lambda _j. } We can view $\rho _0=\rho _{0,j}$ as a function on $\Lambda _j$, with $\rho _0=1$ outside $V_j$. Assume that $\widetilde{\phi }_j\in C^\infty ({\bf R}^{\Lambda _j};{\bf R})$ is a family which satisfies the assumptions of subsection B with a new set of weights $W$ that contains $\rho _0$, such that \eekv{9.69} { (\rho _0\otimes \rho _0)(\phi _j\oplus\psi _j-\widetilde{\phi }_j)''\hbox{ is 2 standard, if } } { \psi _j\in C^\infty ({\bf R}^{\Lambda _j\setminus V_j};{\bf R})\hbox{ and }\psi _j''\hbox{ is 2 standard.} } Here $\rho _0$ is defined as in (9.62) with ${\bf Z}^d$ replaced by $\Lambda _j$. We also assume that $t\widetilde{\phi }_j+(1-t)(\phi _j\oplus \psi _j)$, $0\le t\le 1$, satisfies ($\widetilde{{\rm H}4}$) of section 8. Similarly to (9.67), we get \ekv{9.70} { {\rm Cor\,}_{\widetilde{\phi }_j}(x_\nu ,x_\mu ) -{\rm Cor\,}_{\phi _j}(x_\nu ,x_\mu )={\cal O}(1) h e^{-\theta r_j/4},\ \nu ,\mu \in S_j. } On the other hand we can apply Proposition 9.2 to ${\rm Cor\,}_{\widetilde{\phi }_j}(x_\nu ,x_\mu )$ and get \ekv{9.71} {{\rm Cor\,}_{\phi _j}(x_\nu ,x_\mu )={\cal O}(h)e^{-\theta r_j/4}+h e^{-p_{1,h}^j(\nu -\mu )}q^j(\nu -\mu ;h),\ \nu ,\mu \in S_j, } with $p_{1,h}^j$, $q_j$ as in subsection B. \par (9.67) gives a thermodynamical limit of the correlations, while (9.71) describes their asymptotic behaviour. We now combine the two results, in order to show that $p_{1,h}^j$ has a limit when $j\to \infty $ and that the same thing holds for the terms in the large $\nu $ asymptotic expansion of $q^j$, as well as for the terms in the $h$ asymptotic expansions of these quantities. Let $k\ge j\gg 1$ and take $\mu =0$. For every sufficiently large $C_0\ge 1$, there is a $C_1>0$, such that $$\vert e^{-p_{1,h}^j(\nu )}q^j(\nu ;h)-e^{-p_{1,h}^k(\nu )}q^k(\nu ;h)\vert \le {\cal O}(1) e^{-(p_{1,h}^j(\nu )+r_j/C_1)},$$ for $r_j/C_0^2\le \vert \nu \vert \le r_j/C_0$. This implies that with a new constant $C_1>0$: \ekv{9.72} {\vert 1-e^{p_{1,h}^j(\nu )-p_{1,h}^k(\nu )} {q^k(\nu ;h)\over q^j(\nu ;h)}\vert \le {\cal O}(1)e^{-r_j/C_1},\ {r_j\over C_0^2}\le \vert \nu \vert \le {r_j\over C_0}. } \par Here it will be convenient to write \ekv{9.73} {q^j(\nu ;h)=\vert \nu \vert ^{-{d-1\over 2}}e^{-s^j(\nu ;h)},\ \vert \nu \vert \ge C_0,} where \ekv{9.74} { s^j(\nu ;h)\sim\sum_{-\infty }^0 s^j_\alpha (\nu ;h),\ \vert \nu \vert \to\infty , } uniformly with respect to $h,j$, with $s_\alpha ^j$ positively homogeneous of degree $-\alpha $ in $\nu $, and \ekv{9.75} { s_\alpha ^j\sim\sum_{\beta =0}^\infty s_{\alpha ,\beta }^j(\nu )h^\beta ,\ h\to 0, } with $s_{\alpha ,\beta }^j$ also positively homogeneous of degree $-\alpha $ in $\nu $. All these functions are smooth and uniformly bounded in the appropriate spaces when $j$ varies. From (9.72), we deduce that $$p_{1,h}^j(\nu )-p_{1,h}^k(\nu )+s^j(\nu ;h)-s^k(\nu ;h)={\cal O}(1) e^{-r_j/C_1},\ {r_j\over C_0^2}\le \vert \nu \vert \le{r_j\over C_0}.$$ Using (9.74), we get \ekv{9.76} {p_{1,h}^j(\nu )-p_{1,h}^k(\nu )+\sum_{-N}^0 (s_\alpha ^j(\nu ;h)-s_\alpha (\nu ;h))={\cal O}(r_j^{-(N+1)}),\ {r_j\over C_0^2}\le \vert \nu \vert \le{r_j\over C_0},} where we recall that $C_0$ can be chosen arbitrarily large. For $m\in {\bf R}$, $a:{\bf Z}^d\setminus\{ 0\}\to{\bf R}$, put \ekv{9.77} { (D_ma)(\nu )=a(\nu )-2^{-m}a(2\nu ). } If $a$ is the restriction of a function on ${\bf R}^d\setminus\{0\}$ which is positively homogeneous of degree $n\in {\bf R}$, then $D_ma=(1-2^{n-m})a$, and we observe that the prefactor $1-2^{n-m}$ vanishes precisely for $n=m$. If $-N\le n\le 1$, we apply $$\prod_{m\in\{ -N,..,1\}\setminus\{ n\}}D_m$$ to (9.76) and conclude that \ekv{9.78} {p^j_{1,h}(\nu )-p_{1,h}^k(\nu )={\cal O}(r_j^{-(N+1)})\ \ (n=1)} \ekv{9.79} { s_\alpha ^j(\nu ;h)-s_\alpha ^k(\nu ;h)={\cal O}(r_j^{-(N+1)}),\ \ (n=\alpha ) } for $-N\le \alpha \le 0$, ${r_j\over 2C_N}\le \vert \nu \vert \le {r_j\over C_N}$, $ N\in{\bf Z}^d$, for some $C_N\in [C_0,C_0^2/2]$. \medskip \par\noindent \bf Lemma 9.3. \it Let $\widetilde{\Omega }\subset\subset\Omega \subset{\bf R}^d$ be open, $N_0\in\{ 1,2,..\} $, ${\cal E}\subset ]0,1]$, $0\in\overline{{\cal E}}$. Let $u=u_\epsilon (x)\in C^\infty (\Omega )$, $\epsilon \in{\cal E}$, and assume that \ekv{9.80} {\vert \partial _x^\alpha u_\epsilon (x)\vert \le C_\alpha ,\ x\in\Omega ,\,\epsilon \in{\cal E},} \ekv{9.81} {\vert u_\epsilon (x)\vert \le \epsilon ^{N_0},\ x\in\epsilon {\bf Z}^d\cap\Omega ,} where $C_\alpha $ is independent of $\epsilon $. Then, \ekv{9.82} {\vert \partial _x^\alpha u_\epsilon (x)\vert \le \widetilde{C}_\alpha \epsilon ^{N_0-\vert \alpha \vert },\ x\in\widetilde{\Omega },\, \vert \alpha \vert \le N_0.}\rm\medskip \par\noindent \bf Proof. \rm For $1\le j\le d$, let $e_j$ be the $j$th unit vector in ${\bf R}^d$ and put $$D_{j,\epsilon }u(x)={u(x+\epsilon e_j)-u(x)\over \epsilon }=\int_0^1 (\partial _{x_j}u)(x+t\epsilon e_j)dt.$$ Then \eekv{9.83} { D_{j_1,\epsilon }..D_{j_k,\epsilon }u(x)=\int_0^1..\int_0^1(\partial _{x_{j_1}}..\partial _{x_{j_k}}u)(x+\epsilon t_1e_1+..+\epsilon t_ke_{j_k})dt_1..dt_k } { =\partial _{x_{j_1}}..\partial _{x_{j_k}}u(x)+{\cal O}_k(\epsilon )\sup_{\vert y-x\vert _\infty \le k\epsilon }\max_{\vert \alpha \vert =k+1}\vert \partial ^\alpha u(y)\vert . } Let $\widetilde{\Omega }=:\Omega _{N_0}\subset\subset\Omega _{N_0-1}\subset\subset ...\subset\subset\Omega _1\subset\subset \Omega $ and choose $\epsilon >0$ small enough. We first see that \ekv{9.84} {D_{j_1,\epsilon }..D_{j_k,\epsilon }u(x)={\cal O}(\epsilon ^{N_0-k}),\ x\in\Omega _1\cap\epsilon {\bf Z}^d,\ k\le N_0-1,} then using also (9.83), that $\partial ^\alpha u={\cal O}(\epsilon )$, $\vert \alpha \vert \le N_0-1$. Using (9.83) again, we see that $$\partial ^\alpha u={\cal O}(\epsilon ^2),\ \vert \alpha \vert \le N_0-2,\, x\in\Omega _2.$$ Iterating this argument, we get (9.82). \hfill{$\#$} \medskip \par We apply the lemma to (9.78), (9.79) with $\epsilon ={1\over r_j}$, after the change of variables $\nu =r_j\mu $. Using also the homogeneity, and that $N$ can be chosen arbitrarily large, we conclude that \ekv{9.85} { \partial ^\beta (p^j_{1,h}-p^k_{1,h})={\cal O}_{N,\beta }(r_j^{-N})\vert \nu \vert ^{1-\vert \beta \vert },\ \nu \in{\bf R}^d\setminus\{ 0\} , } \ekv{9.86} { \partial ^\beta (s_\alpha ^j-s_\alpha ^k)={\cal O}_{N,\beta ,\alpha }(r_j^{-N})\vert \nu \vert ^{\alpha -\vert \beta \vert },\ \nu \in{\bf R}^d\setminus\{ 0\} ,} for all multiindices $\beta $, when $k\ge 1$. \par From (9.85) we conclude that there exists $p_{1,h }^\infty (\nu )\in C^\infty ({\bf R}^d\setminus\{ 0\} )$ positively homogeneous of degree 1 in $\nu $, and uniformly bounded in $C^\infty $, when $h$ varies, such that \ekv{9.87} { \partial ^\beta (p_{1,h}^\infty -p_{1,h}^j)={\cal O}_{N,\beta }(r_j^{-N})\vert \nu \vert ^{1-\vert \beta \vert },\ \nu \in{\bf R}^d, } for every multiindex $\beta $. \par Consider the truncated asymptotic expansion of $p_{1,h}^j$: \ekv{9.88} { p_{1,h}^j(\nu )=\sum_{\ell =0}^{M-1}p_{1,\ell}^j(\nu ) h^\ell +R_M^j(\nu ;h), } with \ekv{9.89} { \partial ^\beta R_M^j={\cal O}_{M,\beta }(h^M)\vert \nu \vert ^{1-\vert \beta \vert ,} } uniformly in $j$. Since all functions will be homogeneous for a while, we restrict the attention to a spherical shell in $\nu $, and drop the obvious powers of $\vert \nu \vert $. Using (9.85), we see that $$p_{1,0}^j-p^k_{1,0}={\cal O}(r_j^{-N}+h),$$ for all $N$, implying \ekv{9.90} { p_{1,0}^j-p_{1,0}^k={\cal O}(r_j^{-N}). } Now we can use (9.85,88) once more, to see that \ekv{9.91} { p_{1,1}^j-p_{1,1}^k={\cal O}({r_j^{-2N}\over h}+h), } for every $N$. Choose $h=r_j^{-N}$, to get \ekv{9.92} { p_{1,1}^j-p_{1,1}^k={\cal O}(r_j^{-N}). } Continuing this way, we get \ekv{9.93} { p_{1,\ell}^j-p_{1,\ell}^k={\cal O}_\ell (r_j^{-N}), } for every $N$, and the same estimates hold for $\partial ^\beta (p_{1,\ell}^j-p_{1,\ell}^k)$. Using this and (9.85) in (9.89), we get \ekv{9.94} { R_M^j-R_M^k={\cal O}(r_j^{-N}),\ \forall N. } On the other hand, $R_M^j-R_M^k={\cal O}(h^M)$, by (9.89), so interpolation with (9.94) gives $R_M^j-R_M^k={\cal O}(h^{M-1}r_j^{-N})$ for every $N\ge 0$. Use this estimate with $M$ replaced by $M+1$, as well as (9.93) in the identity \ekv{9.95} { R_M^j-R_M^k=(p_{1,M}^j-p_{1,M}^k)h^M+ (R_{M+1}^j-R_{M+1}^k). } We get \ekv{9.96} {R_M^j-R_M^k={\cal O}(h^M r_j^{-N})\vert \nu \vert ,\ \nu \in{\bf R}^d\setminus\{ 0\},} and the analogous estimate holds for the $\beta $th derivative, with $\vert \nu \vert $ replaced by $\vert \nu \vert ^{1-\vert \beta \vert }$. From (9.93), we get the existence of $p_{1,\ell}^\infty \in C^\infty ({\bf R}^d\setminus\{ 0\} )$, such that \ekv{9.97} { \vert p_{1,\ell}^j-p_{1,\ell}^\infty \vert ={\cal O}_{\ell ,N}(r_j^{-N})\vert \nu \vert , } and similarly for the derivatives. Similarly \ekv{9.98} { \vert R_M^j-R_M^\infty \vert ={\cal O}_{M,N}(h^M r_j^{-N})\vert \nu \vert . } We conclude that \ekv{9.99} { p_{1,h}^\infty (\nu )\sim\sum_{\ell =0}^\infty p_{1,\ell}^\infty (\nu )h^\ell . } \par The same arguments apply to \ekv{9.100} { s_\alpha ^j\sim\sum_{\beta =0}^\infty s_{\alpha ,\beta }^j(\nu ) h^\beta , } and we get \ekv{9.101} { s_\alpha ^j-s_\alpha ^\infty ={\cal O}_{N,\alpha }(r_j^{-N})\vert \nu \vert ^{-\alpha }, } \ekv{9.102} { s_{\alpha ,\beta }^j-s_{\alpha ,\beta }^\infty ={\cal O}_{N,\beta ,\alpha }(r_j^{-N})\vert \nu \vert ^{-\alpha }, } \ekv{9.103} {s_\alpha ^\infty \sim\sum_{\beta =0}^\infty s_{\alpha ,\beta }^\infty (\nu )h^\beta .} \par We combine (9.67), (9.71) to get for $j\le k$: \ekv{9.104} {\vert e^{-p_{1,h}^j(\nu )}q^j(\nu ;h)-e^{-p_{1,h}^k(\nu )}q^k(\nu ;h)\vert \le {\cal O}(1) e^{-\theta r_j/4},\ \vert \nu \vert \le {r_j\over C_0}.} We know that $p_{1,h}^j(\nu )$ is uniformly of the order of $\vert \nu \vert $, so after multiplying with $e^{p_{1,h}^j(\nu )}$, we get (possibly with a new $C_0$): \ekv{9.105} { \vert q^j(\nu ;h)-q^k(\nu ;h)+(1-e^{p_{1,h}^j(\nu )-p_{1,h}^k(\nu )})q^k(\nu ;h)\vert \le {\cal O}(1) e^{-r_j/C_0}, \vert \nu \vert \le {r_j\over C_0}. } Here $q^k={\cal O}((\vert \nu \vert +1)^{-{d-1\over 2}})$ uniformly with respect to $k$ and we have (9.85), so $$(1-e^{p_{1,h}^j-p_{1,h}^k})q^k={\cal O}(r_j^{-N}),\ \vert \nu \vert \le {r_j\over C_0}.$$ Using this in (9.105), we get \ekv{9.106} { \vert q^j(\nu ;h)-q^k(\nu ;h)\vert \le {\cal O}(1) r_j^{-N},\ \vert \nu \vert \le {r_j\over C_0}, } for every $N$. We conclude (cf. (9.87)) that there exists a function $q^\infty (\nu ;h)$, $\nu \in{\bf Z}^d$, $00.} If we had been working on a finite dimensional space, then (A.1,2) would have been a necessary and sufficient condition for the existence of a smooth realvalued function $\phi $ with $\phi ''_{j,k}=\Phi _{j,k}$. Such a function does not in general exist in the infinite dimensional case, but we shall now see how to produce two different finite dimensional versions of such a function. \par Let $U\subset{\bf Z}^d$ be finite. If $x\in{\bf R}^U$, let $\widetilde{x}\in{\bf R}^{{\bf Z}^d}$ be the zero extension of $x$, so that $\widetilde{x}(j)=x(j)$ for $j\in U$, $\widetilde{x}(j)=0$, for $j\in{\bf Z}^d\setminus U$. Then $$\Phi _{U;j,k}(x):=\Phi _{j,k}(\widetilde{x}),\ j,k\in U$$ is a smooth tensor on ${\bf R}^U$ which satisfies (A.1,2) with $j,k,\ell\in U$. Hence there exists a function $\phi _U(x)\in C^\infty ({\bf R}^U;{\bf R})$ with \ekv{10.1} {\phi ''_{U;j,k}(x)=\Phi _{U;j,k}(x),\ x\in{\bf R}^U,\, j,k\in U.} We make $\phi _U$ unique up to a constant, by requiring that \ekv{10.2} {\phi '_U(0)=0.} It is obvious that $\phi ''_U$ is 2-standard, so we have (H1) (section 1) with $\Lambda $ replaced by $U$, and (H2,3) hold. In order to have (H4) of section 1, we introduce the $2$ standard matrix \ekv{10.3} {A(x)=\int_0^1 \Phi (tx) dt,} and assume \eekv{{\rm A}.5} {A(x):\ell^p\to \ell^p\hbox{ has an inverse }B(x):\ell^p\to\ell^p,}{\hbox{which is uniformly bounded for }x\in{\bf R}^{{\bf Z}^d},\, 1\le p\le \infty .} Since $A$ is $2$ standard, we see that $B(x)$ is 2 standard. \par With $U$ as above, we take $x\in{\bf R}^U$ and let as before $\widetilde{x}$ denote the $0$ extension of $x$ to ${\bf R}^{{\bf Z}^d}$. Then we can introduce the 2 standard matrix \ekv{10.4} {A_U(x)=\int_0^1 \phi ''_U(tx)dt=r_U A(\widetilde{x})r_U^*,} where $r_U:{\bf R}^{{\bf Z}^d}\to {\bf R}^U$ is the restriction map. We assume in addition to (A.5), that \eekv{{\rm A}.6} {A_U(x)\hbox{ has an inverse }B_U(x) \hbox{ which}}{\hbox{is uniformly bounded for }x\in{\bf R}^U,\, 1\le p\le \infty , } uniformly for all $U$ in some class of finite $U$ under consideration. With these assumptions we have obtained smooth functions $\phi _U\in C^\infty ({\bf R}^U)$ which satisfy (H.1--4) for $U$ in some class of finite subsets of ${\bf Z}^d$. \par We next do the same with $U$ replaced by a discrete torus $\Lambda =({\bf Z}/L{\bf Z})^d$. If $\lambda \in {\bf Z}^d$, we define $\tau _\lambda x\in{\bf R}^{{\bf Z}^d}$, by $(\tau _\lambda x)(\nu )=x(\nu -\lambda )$. Eventually, we will assume complete translation invariance for $\Phi $: \ekv{{\rm A}.7} {\Phi _{j+\lambda ,k+\lambda }(\tau _\lambda x)=\Phi _{j,k}(x),\ j,k,\lambda \in{\bf Z}^d.} Notice that if $\Phi _{j,k}$ were the Hessian of a smooth function on $\phi \in C^\infty ({\bf R}^{{\bf Z}^d})$ (and the discussion remains valid if we replace ${\bf Z}^d$ by a discrete torus $\Lambda $) then (A.7) would be a consequence of the simpler translation invariance property: \ekv{10.4} {\phi (\tau _\lambda x)=\phi (x).} However, to begin with, we only assume the weaker assumption $$\Phi _{j+\lambda ,k+\lambda }(\tau _\lambda x)=\Phi _{j,k}(x),\ j,k\in{\bf Z}^d,\, \lambda \in L{\bf Z}^d,\eqno ({\rm A}.7)_L$$ for some given $L\in\{ 1,2,..\}$. \par If $x\in{\bf R}^\Lambda $, let $\widetilde{x}\in{\bf R}^{{\bf Z}^d}$ be the corresponding $L{\bf Z}^d$ periodic lift. Replacing $x$ by $\widetilde{x}$ in $({\rm A}.7)_L$, we get \ekv{10.5} {\Phi _{j-\lambda ,k-\lambda }(\widetilde{x})=\Phi _{j,k}(\widetilde{x}),\ \lambda \in L{\bf Z}^d.} If we view $\Phi $ as a matrix, this is equivalent to \ekv{10.6} {\tau _\lambda \circ \Phi (\widetilde{x})=\Phi (\widetilde{x})\circ \tau _\lambda ,\ \lambda \in L{\bf Z}^d,} so $\Phi (\widetilde{x})$ maps $L{\bf Z}^d$ periodic vectors into the same kind of vectors. Hence there is a $\Lambda \times \Lambda $ matrix $\Phi _\Lambda (x)$, defined by \ekv{10.7} { \widetilde{\Phi _\Lambda (x)t}=\Phi (\widetilde{x})\widetilde{t}, } where again the tilde indicates that we take the periodic lift. On the matrix level, we get \ekv{10.8} { \Phi _{\Lambda ;j,k}(x)=\sum_{\widetilde{k}\in\pi _\Lambda ^{-1}(k)}\Phi _{\widetilde{j},\widetilde{k}}( \widetilde{x}), } for any $\widetilde{j}\in \pi _\Lambda ^{-1}(j)$, where $\pi _\Lambda :{\bf Z}^d\to\Lambda $ is the natural projection. Alternatively, we have \ekv{10.9} {\Phi _{\Lambda ;j,k}(x)=\sum_{\widetilde{j}\in\pi _\Lambda ^{-1}(j)}\Phi _{\widetilde{j},\widetilde{k}}( \widetilde{x}),\ \widetilde{k}\in \pi _\Lambda ^{-1}(k),} and $\Phi _{\Lambda ;j,k}$ is symmetric (cf. (A.1)). \par Let us verify the analogue of (A.2). For $j,k,\ell \in\Lambda $ we have \ekv{10.10} { \partial _{x_\ell}\Phi _{\Lambda ;j,k}(x)=\partial _{x_\ell}\sum_{\widetilde{k}\in\pi _\Lambda ^{-1}(k)}\Phi _{\widetilde{j},\widetilde{k}} (\widetilde{x})=\sum_{\widetilde{k}\in\pi _\Lambda ^{-1}(k)}\sum_{\widetilde{\ell}\in\pi _\Lambda ^{-1}(\ell )}\partial _{x_{\widetilde{\ell}}}\Phi _{\widetilde{j},\widetilde{k}}(\widetilde{x} ). } From (A.1,2) we know that $\partial _{x_{\widetilde{\ell}}}\Phi _{\widetilde{j},\widetilde{k}} (\widetilde{x})=\partial _{x_{\widetilde{k}}}\Phi _{\widetilde{j},\widetilde{\ell}} (\widetilde{x})$, so the last expression in (10.10) is symmetric in $\ell,k$ and we get \ekv{10.11} {\partial _{x_{\ell}}\Phi _{\Lambda ;j,k}(x)=\partial _{x_k}\Phi _{\Lambda ;j,\ell}(x).} Using also the symmetry of $\Phi _{\Lambda ;j,k}$, we get the analogue of (A.2). It is now clear that there exists $\phi _\Lambda \in C^\infty ({\bf R}^\Lambda ;{\bf R})$, unique up to a constant, such that \ekv{10.12} {\Phi _{\Lambda ;j,k}(x)=\partial _{x_j}\partial _{x_k}\phi _\Lambda (x),\ \phi _\Lambda '(0)=0.} \par Let us verify that $\phi ''_\Lambda $ is 2 standard. If $k\ge 2$, $t_1,..,t_k\in {\bf C}^\Lambda $, $x\in{\bf R}^\Lambda $, we have \ekv{10.13} {\langle \nabla ^k\phi _\Lambda (x),t_1\otimes ..\otimes t_k\rangle =\langle \nabla ^{k-2}\Phi _\Lambda ,t_1\otimes ..\otimes t_k\rangle =\langle \nabla ^{k_2}\Phi (\widetilde{x}),1_E\widetilde{t}_1\otimes \widetilde{t}_2\otimes ..\otimes \widetilde{t}_k\rangle ,} if $E\subset{\bf Z}^d$ is a fundamental domain for $L{\bf Z}^d$ and $\widetilde{x}$, $\widetilde{t}_j$ denote the periodic lifts. Using that $\Phi $ is 2 standard, we deduce that \ekv{10.14} {\langle \nabla ^k\phi _\Lambda (x),t_1\otimes ..\otimes t_k\rangle ={\cal O}_k(1)\vert t_1\vert _1\vert t_2\vert _\infty ..\vert t_k\vert _\infty .} Here the index 1, can be replaced by any other index in $\{ 1,..,k\}$, and the RHS in (10.14) can therefore be replaced by \ekv{10.15} {{\cal O}_k(1)\vert t_j\vert _1\prod_{{1\le \nu \le k,}\atop {\nu \ne j}}\vert t_\nu \vert _\infty .} By complex interpolation, we get the desired relation \ekv{10.16} {\langle \nabla ^k\phi _\Lambda (x), t_1\otimes ..\otimes t_k\rangle ={\cal O}_k(1) \vert t_1\vert _{p_1}..\vert t_k\vert _{p_k},} uniformly in $x$, $t_j$ and $p_j$, when $1\le p_j\le \infty $, $1={1\over p_1}+..+{1\over p_k}$. \par We next check that $\phi _\Lambda $ satisfies (H.3) (section 1), so we put $x=0$, $\widetilde{x}=0$ and omit these quantities in the formulas. Choose a fundamental domain $E$ and let $(\Psi _{j,k})$ be the block matrix of $\Phi $ with respect to the decomposition $$\ell^2({\bf Z}^d)=\oplus_{k\in{\bf Z}^d}\ell^2(E_k),$$ where $E_k:=kL+E$. Then $\Psi _{j,k}=\Psi _{j-k}$ by slight abuse of notation. Since $\Phi ={\cal O}(1):\ell^p\to\ell^p$, $1\le p\le \infty $, we know that $\Phi _{j,k}$ satisfies the (equivalent) Shur condition $$\sup_j\sum_k \vert \Phi _{j,k}\vert ,\, \sup_k\sum_j\vert \Phi _{j,k}\vert <\infty ,$$ and this implies that $\sum_{k}\Vert \Psi _k\Vert <\infty $, where $\Vert \cdot \Vert $ denotes the norm in ${\cal L}(\ell^2(E),\ell^2(E))$. Now we can write \ekv{10.17} { \langle \Phi _\Lambda t,t\rangle =\sum_k\langle \Psi _kt,t\rangle , } identifying $t\simeq 1_E\widetilde{t}$. We compare this with \eeekv{10.18}{{1\over \# B(0,R)}\sum_{\vert j\vert ,\vert k\vert \le R}\langle \Psi _{j-k}t,t\rangle =} {{1\over \# B(0,R)}\sum_{{\vert j\vert \le (1-\epsilon )R}\atop {k\in {\bf Z}^d}}.. - {1\over \# B(0,R)}\sum_{{\vert j\vert \le (1-\epsilon )R}\atop{\vert k\vert >R}}.. +{1\over \# B(0,R)}\sum_{{(1-\epsilon )R<\vert j\vert \le R}\atop {\vert k\vert \le R}}.. } {={\rm I}+{\rm II}+{\rm III},} where $B(0,R):=\{ j\in{\bf Z}^d;\vert j\vert \le R\}$. Here \ekv{10.19} {{\rm I}={\# B(0,(1-\epsilon )R)\over \# B(0,R)}\sum \langle \Psi _{-k}t,t\rangle = {\# B(0,(1-\epsilon )R)\over \# B(0,R)}\langle \Phi _\Lambda t,t\rangle ,} \ekv{10.20} { \vert {\rm II}\vert \le \sum_{\vert \ell\vert >\epsilon R}\Vert \Psi _\ell \Vert \vert t\vert ^2=o_{\epsilon ,t}(1),\ R\to \infty , } \ekv{10.21} { \vert {\rm III}\vert \le {\# (B(0,R)\setminus B(0,(1-\epsilon )R))\over \# B(0,R)}\sum_k \Vert \Psi _k\Vert \vert t\vert ^2=o_t(1),\ \epsilon \to 0. } It follows that \ekv{10.22} { \langle \Phi _\Lambda t,t\rangle =\lim_{R\to \infty }{1\over \# B(0,R)}\sum_{\vert j\vert ,\vert k\vert \le R}\langle \Psi _{j-k}t,t\rangle =\lim_{R\to\infty }\langle \Phi \widetilde{t}_R,\widetilde{t}_R\rangle , } where $$\widetilde{t}_R={1\over \sqrt{\# B(0,R)}}\sum_{\vert k\vert \le R}1_{E_k}\widetilde{t}.$$ The sum is orthogonal in $\ell^2$, so \ekv{10.23} {\vert \widetilde{t}_R\vert ^2=\vert 1_E\widetilde{t}\vert ^2=\vert t\vert ^2. } On the other hand, by (A.4) we have $\langle \Phi \widetilde{t}_R,\widetilde{t}_R\rangle \ge {1\over {\cal O}(1)}\vert \widetilde{t}_R\vert ^2$, so from this and (10.23,22) we get (H.3) for $\Phi _\Lambda $ with the same constant as in (A.4). \par Next we verify (H.4) for $\phi _\Lambda $. For that we notice that we can define a gradient $\phi '(x)$ at $x\in{\bf R}^{{\bf Z}^d}$ if $\vert x\vert _\infty <\infty $, by \ekv{10.24} {\phi '(x)=\int_0^1\Phi (tx)xdt=A(x)x,} or more explicitly by \ekv{10.25} {\phi _j'(x)=\sum_k \int_0^1\Phi _{j,k}(tx)x_k dt,} and we verify that $$\partial _{x_\ell}\phi _j'(x)=\Phi _{j,\ell } (x)\ \ \ (=\partial _{x_j}\phi '_\ell (x)\, )$$ by a straight forward computation: $$\eqalign{&\partial _{x_\ell}\phi _j'(x)=\sum_k\int_0^1 (\partial _{x_\ell}\Phi _{j,k})(tx)tx_k dt+\int_0^1 \Phi _{j,\ell }(tx) dt=\cr & \int_0^1\sum_k (\partial _{x_k}\Phi _{j,\ell })(tx) tx_k dt +\int \Phi _{j,\ell}(tx) dt=\int_0^1 (t\partial _t+1)(\Phi _{j,\ell}(tx))dt=\Phi _{j,\ell}(x).}$$ Put $A_\Lambda (x)=\int_0^1 \Phi _\Lambda (tx) dt$ so that \ekv{10.26} {\phi '_\Lambda (x)=A_\Lambda (x)x.} The relation between $A_\Lambda (x)$ and $A(\widetilde{x})$ is the same as between $\Phi _\Lambda $ and $\Phi $: \ekv{10.27} {\widetilde{A_\Lambda (x)t}=A(\widetilde{x})\widetilde{t}.} Since $A(\widetilde{x})$ is uniformly invertible in $\ell^\infty ({\bf Z}^d)$ it has the same property on the invariant subspace of $L{\bf Z}^d$ periodic vectors. This means that $A_\Lambda (x):\ell^\infty (\Lambda )\to\ell^\infty (\Lambda )$ has a uniformly bounded inverse. Since $A_\Lambda $ is symmetric, we have the same property on $\ell^1(\Lambda )$ and by interpolation on $\ell^p (\Lambda )$. \par Assume that for every $C_0>0$ \eekv{{\rm A}.8} {\rho ^{-1}\Phi (x)\rho \hbox{ is 2 standard, uniformly for every $\rho :{\bf Z}^d\to ]0,\infty [$ of the }} {\hbox{form $\rho (j)=e^{r(j)}$ with $r:{\bf R}^d\to {\bf R}$ of Lipschitz class with $\vert \nabla r\vert \le C_0$ a.e.}} Let us then verify ($\widetilde{{\rm H}.1}$) (section 7) for $\phi _U$ and $\phi _\Lambda $ with a suitable class of weights $W$. In the first case we let $W=W_U$ be the set of weights of the form ${\rho _\vert }_U$ with $\rho $ as in (A.8) for an arbitrary but fixed $C_0>0$. Then ($\widetilde{{\rm H}.1})$ holds for $\phi _U$. In the second case, we let $W_\Lambda $ be the set of $\rho (j)=e^{r(j)}$ with $r:({\bf R}/L{\bf Z})^d\to{\bf R}$ of Lipschitz class with $\vert \nabla r\vert \le C_0$ a.e., and again $C_0>0$ is arbitrary but fixed. Let $\widetilde{\rho }=e^{\widetilde{r}}={\bf Z}^d\to ]0,\infty [$ be the corresponding periodic lift. Then we have the analogue of (10.13): \ekv{10.28}{\langle \nabla ^{k-2}\rho ^{-1}\Phi _\Lambda (x)\rho ,t_1\otimes ..\otimes t_k\rangle =\langle \nabla ^{k-2}\widetilde{\rho }^{-1}\Phi (\widetilde{x})\widetilde{\rho }, 1_E\widetilde{t}_1\otimes \widetilde{t}_2\otimes ..\otimes \widetilde{t}_k\rangle ,\ k\ge 2,} where again the index 1 can be replaced by any other index in $\{ 1,..,k\}$. As before, we see that $\rho ^{-1}\Phi _\Lambda (x)\rho $ is 2 standard uniformly for $\rho \in W_\Lambda $, so $\phi _\Lambda $ satisfies ($\widetilde{{\rm H}.1}$) with $W_\Lambda $ as above, for every fixed $C_0>0$. \par We want to apply the discussion of section 9, and we assume (A.1--8) from now on. Since we now have adopted (A.7), completely, we see that $\phi _\Lambda (0)$ is a convolution matrix. We assume that $\Phi (0)$ is ferromagnetic in the sense that \ekv{{\rm A}.9}{\Phi _{j,k}(0)\le 0,\ j\ne k.} Then (H.9) holds for $\phi _\Lambda $ and we have \ekv{10.29} {\Phi (0)= I-\widetilde{v}_0*,\ \ \phi_\Lambda ''(0)=(1+o(1))I _\Lambda -v_0*,\ L\to\infty } and $v_0$ and $\widetilde{v}_0$ are related by (9.12) and $v_0(\nu )$ is even $\ge 0$ and vanishes for $\nu =0$. Again we chose the constant 1 for simplicity, by a dilation in $h$ we can always reduce ourselves to that case. \par Assume that there exists a finite set $K\subset {\bf Z}^d$ such that \ekv{{\rm A}.10} {\widetilde{v}_0(j)>0,\, j\in K,\ {\rm Gr\,}(K)={\bf Z}^d,} where ${\rm Gr\,}(K)$ denotes the smallest subgroup of ${\bf Z}^d$ which contains $K$. This is precisely the assumption (H.10) for the functions $\phi _\Lambda $. Since $\widetilde{v}_0(k)={\cal O}_{C_0}(1)e^{-C_0\vert k\vert }$ for every $C_0>0$, the function $F_{\widetilde{v}_0}(\eta )$ in (9.14) is well defined on ${\bf R}^d$ and from (A.10) and the fact that $\widetilde{v}_0>0$, we see that $\lim_{\vert \eta \vert \to \infty }F_{\widetilde{v}_0}(\eta )=+\infty $. We then have(H.11) and (9.17H) for suitable sets $\Omega $, $\widetilde{\Omega }$. Then the whole discussion of part B of section 9 applies and we have Proposition 9.2 for $\phi _\Lambda $ (when $L$ is large enough). \par We next look at part C of section 9, where we shall take $V_j=U_j\nearrow {\bf Z}^d$, $j\to \infty $ with $U_j$ bounded. For $U\subset {\bf Z}^d$ finite, define $\rho _0=\rho _{0,U}:{\bf Z}^d\to ]0,\infty [$ as in (9.62): \ekv{10.30} { \rho _0(\nu )=\exp \theta {\rm dist\,}(\nu ,{\bf Z}^d\setminus U), } for some fixed $\theta >0$. In order to have (H12) (which implies (H5) in the discussion in section 9) we need an assumption: \ekv{{\rm A}.11} {\big( \rho _{0,U}(j)\rho _{0,U}(k)(\Phi _{j,k}(1_Ux)-\Phi _{j,k}(x))\big)_{j,k\in U}\hbox{ is 2 standard.}} Here $1_U$ is the characteristic function of $U$, so $(1_Ux)(j)$ is equal to $x(j)$ when $j\in U$ and equal to $0$ when $j\in{\bf Z}^d\setminus U$. We observe that (A.11) and (A.8) follow from (A.1--3) and the following finite range condition: \ekv{{\rm A.fr}} {\exists C_0,\hbox{ such that }\Phi _{j,k}(x)=0\hbox{ for }\vert j-k\vert >C_0.} In fact, if (A.fr) holds, then $\Phi _{j_1,j_2,..,j_m}:=\partial _{x_{j_3}}..\partial _{x_{j_m}}\Phi $ vanishes if $\vert j_\nu -j_\mu \vert >C_0$ for some $\nu ,\mu $, and (A.3) is equivalent to the statement that $\Phi _{j_1,..,j_m}(x)={\cal O}_m(1)$ for $m=2,3,..$. Similarly (A.8) is equivalent to $${\rho (j_2)\over\rho (j_1)}\Phi _{j_1,..,j_m}(x)={\cal O}_m(j),$$ for $\rho $ as in (A.8). Since $(\rho (j_2)/\rho (j_1))\Phi _{j_1,..,j_m}={\cal O}(1)\Phi _{j_1,..,j_m}$, we see that (A.8) follows from (A.3) and (A.fr). Moreover, $\partial _{x_\ell}\Phi _{j,k}=\partial _{x_j}\Phi _{\ell ,k}=..$ vanishes if $\vert j-\ell\vert $ or $\vert k-\ell\vert $ is $>C_0$ and consequently the expression (A.11) vanishes as soon as ${\rm dist\,}(j,{\bf Z}^d\setminus U)$ or ${\rm dist\,}(k,{\bf Z}^d\setminus U)$ is larger than $C_0$. This means that we can replace $\rho _{0,U}$ by some uniformly bounded functions $\widetilde{\rho }_{0,U}$ without changing the expression in (A.11), and (A.11) then follows from the 2 standardness of $\Phi _{j,k}(1_Ux)$ and of $\Phi _{j,k}(x)$. More generally (A.11) holds if we assume that there is a $C_0>0$ such that $\Phi _{j,k}(x)=\Phi _{j,k}(0)$ whenever $\vert j-k\vert >C_0$. Indeed, we again obtain that $\partial _{\ell}\Phi _{j,k}=0$ if $\vert k-\ell\vert >C_0$ or $\vert j-\ell\vert >C_0$. We add (A.11) to our assumptions from now on, and verify (H.12). If $U\subset\widetilde{U} \subset {\bf Z}^d$ are finite, then $(\rho _{0,U}\otimes \rho _{0,U})(\phi _U\oplus 0-\phi _{\widetilde{U}})''$ is equal to the tensor \ekv{10.31} { \rho _0(j)\rho _0(k)(\Phi _{j,k}(1_Ux)(1_Uj)(1_Uk)-\Phi _{j,k}(1_{\widetilde{U}}x)),\ j,k\in \widetilde{U}, } and we split this tensor into four, according to the cases $j\in U$ or not, $k\in U$ or not. In the three cases where at least one of $j,k$ is in $\widetilde{U}\setminus U$, we know from (A.8) that both $\rho _0(j)\rho _0(k)\Phi _{j,k}(1_Ux)$ and $\rho _0(j)\rho _0(k)\Phi _{j,k}(1_{\widetilde{U}}x)$ are 2 standard, and in the remaining case when both $j$ and $k$ belong to $U$, the tensor (10.31) is 2 standard by (A.11). This means that we have verified (H.12) for $\phi _j=\phi _{U_j}$. \par We have already checked ($\widetilde{\rm H.1}$) (section 7) with the set of weights $W$ above and we next look at ($\widetilde{\rm H.4}$) (section 8), that we need to check for the new and smaller set of weights in (9.63), of the form $\rho =e^r$, $\vert \nabla r\vert \le 2\theta $ a.e. Using the observation after the statement of ($\widetilde{\rm H.4}$), we only need to check that the inverse $B_U(x)$ of $A_U(x)$ (which exists and is uniformly bounded by (A.6)) remains uniformly bounded after conjugation with a weight $\rho $ as in (9.63), if $\theta >0$ is sufficiently small. However, using the Shur class remark, we see that $\Vert \rho ^{-1}A_U\rho -A_U\Vert _{{\cal L}(\ell^p,\ell^p)}$, $1\le p\le \infty $ is as small as we like if $\theta $ is small enough (but independent of $U$), and consequently $(\rho ^{-1}A_U\rho )^{-1}$ is uniformly bounded, so we have checked ($\widetilde{{\rm H}.4}$) for $\phi _U$. \par We also need $(\widetilde{\rm H.4})$ for $\phi _{j,k,t}$ given after (9.66), so that \eekv{10.32} {\hskip -1cm (\phi ''_{j,k,t}(x)_{\nu ,\mu }=t\Phi _{\nu ,\mu }(1_{U_k}x)+(1-t)(1_{U_j}(\nu )1_{U_j}(\mu )\Phi _{\nu ,\mu} (1_{U_j}x)+1_{U_k\setminus U_j}(\mu )\delta _{\nu ,\mu }),}{\hskip 7cm\ \nu ,\mu \in U_k,\, x\in{\bf R}^{U_k}.} The corresponding matrix $A$ becomes $$ A_{j,k,t}(x)=tA_{U_k}(x)+(1-t)(1_{U_j} A_{U_j}(x)1_{U_j} +1_{U_k\setminus U_j}),$$ and we assume in analogy with (A.6) that \eekv{{\rm A}.12} {A_{j,k,t}(x)\hbox{ has an inverse }B_{j,k,t}(x):\ell^p\to \ell^p,\hbox{ which}} {\hbox{is uniformly bounded for }x\in{\bf R}^{U_k},\, 1\le p\le \infty .} Using the Shur class point of view, we see as before that $(\widetilde{\rm H4})$ is fulfilled with $W$ as in (9.63). \medskip \par\noindent \bf Remark 10.1. \rm Using the maximum principle of [S4] we can get a simple condition which implies (A.5), (A.6), (A.12) and the similar condition (A.13) below. Assume for $A(x)=\int_0^1 \Phi (sx)ds$: \eekv{{\rm A.mp}} {\exists \epsilon _0>0\hbox{ such that for every $x\in {\bf R}^{{\bf Z}^d}$, $A(x)$ satisfies (${\rm mp\,}\epsilon _0$): If }} {\hbox{$t\in \ell^1({\bf Z}^d)$, $s\in \ell^\infty ({\bf Z}^d)$, and $\langle t,s\rangle =\vert t\vert _1\vert s\vert _\infty $, then $\langle A(x)t,s\rangle \ge \epsilon _0 \vert t\vert _1\vert s\vert _\infty $.}} It is easy to check (first for $p=1,\infty $ and then by interpolation for intermdiate values of $p$) that $A(x):\ell^p\to\ell^p$ has a uniformly bounded inverse $B(x)$, so (A.mp) implies (A.5). Moreover, if $A(x)$ satifies (mp$\epsilon _0$), so does $1_UA(x)1_U^*$ (as a $U\times U$ matrix), so we get (A.6). Finally, the set of matrices which satisfy (mp$\epsilon _0$) is convex, so $A_{j,k,t}(x)$ will also satisfy (mp$\epsilon _0$) and consequently we will have (A.12). \medskip \par So if we add the assumption (A.12), or replace (A.5,6) by (A.mp), then ($\widetilde{\rm H}4)$ holds for $\phi _{j,k,t}$, and we get (9.67). \par Now let $\Lambda =\Lambda _j=({\bf Z}/L_j{\bf Z})^d$ be a sequence of discrete tori with \ekv{10.33} {U_j\subset [-{L_j\over 4},{L_j\over 4}]^d,} so that we can view $U_j$ as a subset of $\Lambda _j$ in the natural way. Let $\widetilde{\phi }_j=\phi _{\Lambda _j}$. We need to check (9.69) with $\rho _0(\nu )=\rho _{0,j}(\nu )=\exp \theta {\rm dist\,}(\nu ,\Lambda _j\setminus U_j)$, $\nu \in\Lambda _j$. As before, we see that it suffices to check the 2 standardness of \ekv{10.34} {\rho _0(\nu )\rho _0(\mu )(\Phi _{\nu ,\mu }(1_{U_j}x)-\Phi _{\Lambda _j,\nu ,\mu }(x)),\ \nu ,\mu \in U_j.} Recall that $U_j$ is viewed as a subset of $\Lambda _j$ and let $\widetilde{x}$ denote the $L_j{\bf Z}^d$ periodic lift of $x$. Write (10.34) as the difference of the following two expressions: \ekv{10.35} {\rho _0(\nu )\rho _0(\mu )(\Phi _{\nu ,\mu }(1_{U_j}\widetilde{x})-\Phi _{\nu ,\mu }(\widetilde{x})),} and \ekv{10.36} {\rho _0(\nu )\rho _0(\mu )\sum_{0\ne \alpha \in{\bf Z}^d}\Phi _{\nu ,\mu +L_j\alpha }(\widetilde{x}).} Thanks to (A.8), the last expression is 2 standard if we replace $\widetilde{x}$ by a general $x\in{\bf R}^{{\bf Z}^d}$. The same holds for (10.35) by (A.11). As with $\Phi _\Lambda $, we then see that (10.35), (10.36) are 2 standard, and that completes the verification of (9.69). \par We finally need to check that $t\phi _{\Lambda _j}+(1-t)(\phi _{U_j}\oplus \psi _j)$ satisfies ($\widetilde{\rm H.4}$) of section 8, with $\psi _j(x)=\sum_{\nu \in \Lambda _j\setminus U_j}{1\over 2}x_\nu ^2$, and as before, we see that we only need the uniform invertibility in ${\cal L}(\ell^p,\ell^p)$ of \ekv{10.37} {\widetilde{A}_{j,k}=tA_{\Lambda _j}(x)+(1-t)(1_{U_j}A_{U_j}(1_{U_j}x) 1_{U_j}+1_{\Lambda _j\setminus U_j}),\ x\in{\bf R}^{\Lambda _j}.} Assume \ekv{{\rm A}.13} {\widetilde{A}_{j,t}(x):\ell^p\to\ell^p \hbox{ is uniformly invertible as in (A.12).}} Fortunately (A.13) is also a consequence of (A.mp). To see that, it suffices to verify that with $\Lambda =\Lambda _j$, $A_{\Lambda }(x)$ has the property (mp$\epsilon _0$) as we shall now do: Let $t\in\ell^1(\Lambda )$, $s\in\ell^\infty (\Lambda )$ satisfy $\langle t,s\rangle =\vert t\vert _1\vert s\vert _\infty $. We have the obvious analogue of (10.22): \ekv{10.38} {\langle A_\Lambda t,s\rangle =\lim_{R\to\infty }\langle A\widetilde{t}_R,\widetilde{s}_R\rangle ,} with $\widetilde{t}_R$, $\widetilde{s}_R$ defined as after (10.22). Here $\langle \widetilde{t}_R,\widetilde{s}_R\rangle _{\ell^2({\bf Z}^d)}=\langle t,s\rangle _{\ell^2(\Lambda )}$, while $$\vert \widetilde{t}_R\vert _1\vert \widetilde{s}_R\vert _\infty =\vert t\vert _1\vert s\vert _\infty =\langle t,s\rangle =\langle \widetilde{t}_R,\widetilde{s}_R\rangle ,$$ so (A.mp) implies that $\langle A\widetilde{t}_R,\widetilde{s}_R\rangle \ge \epsilon _0\vert \widetilde{t}_R\vert _1\vert \widetilde{s}_R\vert _\infty =\epsilon _0\vert t\vert _1\vert s\vert _\infty .$ Hence by (10.38), $\langle A_\Lambda t,s\rangle \ge \epsilon _0\vert t\vert _1\vert s\vert _\infty $, and we have checked that $A_\Lambda $ satisfies (mp$\epsilon _0$) and hence that we have (A.13) when (A.mp) holds. Summing up, we have verified that the assumptions (A.1--13) imply the results of part C in section 9, as will be restated in the main theorem below. We have also seen that the more explicit conditions (A.fr) and (A.mp) permit to reduce the number of conditions and to simplify them in the sense that they only concern $\Phi _{j,k}$ and not the particular choice of sequences $U_j$ and $\Lambda _j$. Indeed we have verified the implications: $$\hbox{(A.1--3), (A.fr)$\Rightarrow$ (A.8), (A.11),}$$ $$\hbox{(A.1--3), (A.mp)$\Rightarrow$ (A.5,6,12),}$$ $$\hbox{(A.1--3), (A.7), (A.mp)$\Rightarrow$ (A.13).}$$ Also notice that (A.4) follows from (A.mp). Especially (A.1--3,7,9,10,fr,mp) imply (A.1--13). \medskip \par\noindent \bf Theorem 10.2. \it Let $\Phi _{j,k}(x)\in C^\infty ({\bf R}^{{\bf Z}^d})$ satisfy (A.1--5,7--10) and define \break $\phi _U\in C^\infty ({\bf R}^U;{\bf R})$, $\phi _\Lambda \in C^\infty ({\bf R}^\Lambda ;{\bf R})$ as above, when $U\subset {\bf Z}^d$ is finite and $\Lambda =({\bf Z}/L{\bf Z})^d$ is a discrete torus. Let $U_j\subset {\bf Z}^d$, $j=1,2,..$ be an increasing sequence of finite subsets with $0\in U_1$, and assume that $r_j:={\rm dist\,}(0,{\bf Z}^d\setminus U_j)\to \infty $, $j\to \infty $. Choose $\Lambda _j=({\bf Z}/L_j{\bf Z})^d$ with $U_j=[-{L_j\over 4},{L_j\over 4}]$. Assume also that (A.6, 11--13) hold and recall that the assumptions (A.1--3,7,9,10), (A.fr), (A.mp) imply (A.1--13). Then there exist $C_0\ge 1$, $j_0\in{\bf N}$, $\theta >0$, $h_0>0$, such that for $j\ge j_0$, $0