This is a multi-part message in MIME format. ---------------0107060831144 Content-Type: text/plain; name="01-247.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="01-247.keywords" Bootstrap Percolation ---------------0107060831144 Content-Type: application/x-tex; name="boot18.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="boot18.tex" \documentclass[twoside,reqno,12pt,notitlepage]{amsproc} %\usepackage[dvips,final]{graphicx} %sostituire pctex32 con dvips %\usepackage{psfrag} \usepackage{hyperref} %\linespread{1.6} \marginparwidth 70pt \headheight 30 pt \oddsidemargin = 0pt \evensidemargin = 30pt \textwidth= 430pt %\setlength{\baselineskip}{30pt} \parindent=0pt \newtheorem{thm}{\bf Theorem}[section] \newtheorem{prop}[thm]{\bf Proposition} \newtheorem{defin}[thm]{\textsl{\bf Definition}{}} \newtheorem{lemma}[thm]{\bf Lemma} \newtheorem{ans}[thm]{\bf Ansatz} \newenvironment{dimo}{\begin{proof}[{\bf {Proof}}]}{\end{proof}} \numberwithin{equation}{section} \usepackage{latexsym} \usepackage{amsfonts,amssymb} \usepackage{enumerate} \smallskipamount=0.3truecm \medskipamount=0.6truecm \bigskipamount=1truecm %%%%%%%%%%%%%% miedef % %%%%%%%%%%%%%%grafici %%%%%%%%%%%%%%%%%%%%%%%%% GRECO \let\a=\alpha \let\b=\beta \let\c=\chi \let\d=\delta \let\e=\varepsilon \let\f=\varphi \let\g=\gamma \let\h=\eta \let\k=\kappa \let\l=\lambda \let\m=\mu \let\n=\nu \let\o=\omega \let\p=\pi \let\ph=\varphi \let\r=\rho \let\s=\sigma \let\t=\tau \let\th=\vartheta \let\y=\upsilon \let\x=\xi \let\z=\zeta \let\D=\Delta \let\F=\Phi \let\G=\Gamma \let\L=\Lambda \let\Th=\Theta \let\O=\Omega %\let\P=\Pi \let\Ps=\Psi \let\Si=\Sigma \let\X=\Xi \let\Y=\Upsilon %%%%%%%%%%%%%%%%%%%%%%% CALLIGRAFICHE % \def\cA{{\cal A}} \def\cB{{\cal B}} \def\cC{{\cal C}} \def\cD{{\cal D}} \def\cE{{\cal E}} \def\cF{{\cal F}} \def\cG{{\cal G}} \def\cH{{\cal H}} \def\cI{{\cal I}} \def\cJ{{\cal J}} \def\cK{{\cal K}} \def\cL{{\cal L}} \def\cM{{\cal M}} \def\cN{{\cal N}} \def\cO{{\cal O}} \def\cP{{\cal P}} \def\cQ{{\cal Q}} \def\cR{{\cal R}} \def\cS{{\cal S}} \def\cT{{\cal T}} \def\cU{{\cal U}} \def\cV{{\cal V}} \def\cW{{\cal W}} \def\cX{{\cal X}} \def\cY{{\cal Y}} \def\cZ{{\cal Z}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% mie \def\P{{\Bbb P}} \def\R{{\Bbb R}} \def\E{{\Bbb E}} \def\N{{\Bbb N}} \def\fracscript#1#2{\frac{\scriptstyle #1}{\scriptstyle #2}} \def\compo{\circ} \def\mm#1#2{m_-(#1,#2,p)} % #1=d #2=\ell \def\LL#1#2{{\L^{#1}(#2)}} % #1=d #2=lato \def\Lcrit#1#2{\LL{#1}{\mm{#1}{#2}}} % #1=d #2=\ell \def\eexp#1{\exp^{ \compo #1}} % #1= iterazioni \def\esp{\frac{1}{d-\ell+1}} \def\dl{{d,\ell}} \def\mioto{\stackrel{p \to 0}{\longrightarrow}} \def\costa{\b} \def\ca{\a} \def\costp{\ca_{+}} \def\costm{\ca_{-}} \def\costb{\f} \def\costc{\g} % #1=p \def\defmmeno#1#2{\eexp{(#2-2)} % #1=d #2=\ell \left( {\costa_-(#1,#2)}{ p^{{-\frac{\scriptstyle 1} {\scriptstyle #1-#2+1}}}}\right)} \def\defmpiu#1#2{\eexp{(#2-2)} % #1=d #2=\ell \left( {\costa_+(#1,#2)}{ p^{{-\frac{\scriptstyle 1}{\scriptstyle #1-#2+1}}}}\right)} \def\defn#1#2{\eexp{(#2-3)} \left( {\costb(#1,#2)}{ % #1=d #2=\ell p^{{-\frac{\scriptstyle 1}{\scriptstyle #1-#2+1}}}}\right)} \def\boott#1#2{X^{#1}_#2} % #1=d #2=vol \def\boot#1#2#3{\boott{#1}{#2}(#3)} % #1=d #2=vol #3=x \def\YY#1#2#3{Y^#1(#3)} % #1=d #2=vol #3=x \def\ZZ#1#2#3{Z^#1(#3)} % #1=d #2=vol #3=x \def\iniz#1#2#3{X^#1_{#2,0}(#3)} % #1=d #2=vol #3=x \def\ddim{$d$-dimensional } \def\EEa#1#2#3{{#1 \conx{\dl}#2 \ \text{\rm in } {#3}}} % #1=x #2=y #3=vol \def\EE#1#2#3#4{\EEa{#1 }{#2}{\LL{#3}{#4}}} % #1=x #2=y #3=d #4=\ell \def\conk#1{\stackrel{Z}{\longleftrightarrow}} \def\conx#1{\stackrel{\; X^{#1}}{\longleftrightarrow}} \def\longconx#1{\stackrel{\; X^{#1}}{\longleftarrow\!\longrightarrow}} \def\EEEa#1#2#3{{#1 {\conk{#3}} #2 \ \hbox{{ in }} {#3}}} % #1=x #2=y #3=gamma \def\EEE#1#2#3{\EEEa{#1}{#2}{\slice{#3}}} % #1=x #2=y #3=k \def\norma#1{||#1||_\infty} % #1=x \def\ux{\underline x} \def\uy{\underline y} \def\bx{\overline x} \def\by{\overline y} \def\xd#1#2{(#1,#2)} % #1=\ux #2=x \def\slice#1{{T_{{#1}}}} % #1=k \def\cc{\costc} \def\tonda#1{\left( #1 \right)} \def\quadra#1{\left[ #1 \right]} \def\graffa#1{\Big\{ #1 \Big\}} \def\nota#1{\marginpar{\tiny #1}} %%%%%%%%%%% compatibilita` \def\acapo{\par\noindent} \def\Bbb{\mathbb} \def\cal{\mathcal} %\def\Eq{\label} \def\QED{\par\rightline{$\mbox$}} \def\Z{\Bbb Z} %\def\Bbb#1{\fam\msbfam\relax#1} %%%%%%%%%%%%%% \title[\today]{{The threshold regime of finite volume bootstrap percolation.}} \author[R. Cerf, F. Manzo]{R. Cerf\footnote{Universit\'e Paris Sud, Orsay, France.}, F. Manzo\footnote{Universit\'e Paris Sud, Orsay, France and Technische Universit\"at, Berlin, Germany.}.\footnote{Work partially supported by the European network ``Stochastic Analysis and Its Applications'' ERB-FMRX-CT96-0075} \today} \begin{document} \bigskip \maketitle \begin{abstract} We prove that the threshold regime for bootstrap percolation in a $d$-dimensional box of diameter $L$ with parameters $p$ and $\ell$, where $3 \le \ell \le d$, is $L \sim \eexp{(\ell -1)} (C p^{-1/(d-\ell +1)})$, where $\eexp{(\ell -1)}$ is the exponential iterated $\ell-1 $ times and $C$ is bounded from above and from below by two positive constants depending on $d$, $\ell$ only. \end{abstract} %\vfill %\eject \section{ \label{Section 1.} Introduction. } \ We consider the {\it bootstrap percolation} model, with initial occupation density $p$ and parameter $\ell$, in a finite set $\G \subset \Z^d$. More precisely, each site $x$ of $\G \subset \Z^d $ is initially independently occupied with probability $p$ and empty with probability $1-p$. Afterwards, we increase deterministically the set of occupied sites in $\G$ with the help of the following rule, until exhaustion: any site with at least $\ell$ occupied nearest neighbors in $\G$ is occupied. For a discussion on the physical relevance of this model, we refer to \cite{[AL]}; for a nice review paper on bootstrap percolation, see \cite{[Ad]}. We say that a set $\G$ is {\it internally spanned} if all its sites are occupied in the final configuration. The basic question we are interested in is whether or not $\LL{d}{L}$ is internally spanned, where $\LL{d}{L}$ is the $d$-dimensional cubic box of diameter $L$. Let us denote the probability of this event by $$ R(L,p,\dl):= \P \left( \lower 6pt \vbox{ \hbox{\text{the box} $\LL{d}{L}\text{ is internally spanned by the bootstrap }$} \hbox{$ \text{percolation process in } \LL{d}{L} \text{ with parameters } p\text{ and }\ell $}} \right) $$ We focus on the behavior of $R(L,p,\dl)$ when $L$ goes to infinity and $p$ goes to zero. In the case $\ell >d $ we have that $$ \lim_{(L,p) \to (\infty,0)} R(L,p,\dl) =0. $$ Indeed, the presence of a small empty cubic region in the initial configuration precludes the complete filling of $\LL{d}{L}$. More interesting is the case $\ell \le d$. Obviously, for $L$ fixed and $p$ very small the initial configuration will be completely empty with high probability, hence $ \lim_{p \to 0} R(L,p,\dl) =0. $ On the other hand, from the much less obvious results of van Enter \cite{[vE]} and Schonmann \cite{[S]}, we know that for $p$ fixed and $\ell \le d$, $ \lim_{L \to \infty} R(L,p,\dl) =1. $ Therefore, we see that $ \lim_{L \to \infty} \lim_{p \to 0} R(L,p,\dl) =0, $ while $ \lim_{p \to 0} \lim_{L \to \infty} R(L,p,\dl) =1. $ These different limiting behaviors indicate the occurrence of an interesting phenomenon: if we send simultaneously $L \to \infty$ and $p \to 0$, the limit of $R(L,p,\dl) $ will depend on the relative speeds of these convergences, i.e. if $p$ goes extremely quickly (respectively slowly) to $0$ compared to the way $L$ goes to $\infty$, then $R(L,p,\dl) $ will converge to $0$ (respectively $1$). A natural problem is to describe precisely each regime and the threshold between them. In \cite{[AL]}, Aizenman and Lebowitz handled the case $\ell=2$, $d \ge 2$. The threshold regime is $$ L \sim \exp \tonda{\text{const }p^{-\frac{1}{d-1}}}. $$ In \cite{[CeCi]}, Cerf and Cirillo analyzed the case $d=\ell=3$, for which the threshold regime turned out to be $$ L \sim \exp \exp \tonda{\text{const }p^{-1}}. $$ We deal here with the general case $2<\ell \le d$. While the proof of the upper bound on $L$ derives directly from an idea of \cite{[ADE]} and the results of \cite{[S]}, the proof of the lower bound is obtained by using induction on the parameters $(\ell,d)$: by using the technique introduced in \cite{[CeCi]}, we reduce the estimate of the spanning probability for the model $(\ell,d)$ to the spanning probability for the model $(\ell-1,d-1)$. The base of the induction is the Aizenman-Lebowitz case $d \ge \ell=2$. A very challenging and interesting open problem is to decide whether a sharp constant can be put in the exponentials to separate the two regimes. Similar interesting questions can be raised in anisotropic models, as considered for instance in \cite{[M]}. %[omissis organizzazione del'articolo]\ \section{ \label{Section 2.} Basic notation. } \ %[omissis]\ For $t \in \R$, $n \in \N$, we denote by $\eexp{n}(t) $ the exponential {\it iterated $n$ times} of $t$: we set $\eexp{0}(t) := t$ and $\eexp{(n+1)}(t) := \exp ( \eexp{n}(t))$. By $\LL{d}{l}$ we denote the \ddim hypercube with diameter $l$ centered at $0$. Let us give some definitions related to {\it site percolation} (see \cite{G}). On a finite set $\G \subset \Z^d$, let us consider a random configuration $\o \in \{0,1\}^\G$ obtained by occupying (namely, by setting $\o(x)=1$) the sites with the product probability measure $\P_d^p$ with density $p$. We denote by %\begin{equation*} % \label{eq:defprob} $\P_d^p \tonda{\cE}$ %\end{equation*} the probability of the event $\cE$ ($\cE$ is a set of configurations in $\{0,1\}^\G$). %with respect to the product measure with %probability $p$ on $\G \subset \Z^d$. We say that a configuration $\o$ is {\it larger} than a configuration $\o'$ if the set of the occupied sites in the former contains the set of the occupied sites in the latter. An event $\cE$ is called {\it increasing} if %for any pair of configurations %$\o, \o' \in \{0,1\}^\G$ %such that $\o'$ is larger than $\o$ and $\o \in \cE$, %we have $\o' \in \cE$. for any configuration $\o \in \cE$, all configurations $\o' > \o$ are in $\cE$. Our main object of investigation is the following {\it bootstrap} process, defined as a function of a site percolation configuration. On a finite set $\G \subset \Z^d$, let us consider a random initial configuration obtained by occupying the sites with a product measure with probability $p$. We update this initial configuration by using iteratively the following deterministic rule: \begin{enumerate} \item we occupy every empty site with at least $\ell$ occupied nearest neighbors. \item we leave all other sites unchanged. \end{enumerate} Since $\G$ is finite, and the updating procedure cannot empty occupied sites, this procedure stops after a finite number of steps. We denote by $\boott{{d,\ell}}{\G}$ the final configuration of the \ddim bootstrap process in the set $\G$. Thus $\boott{{d,\ell}}{\G}$ is a random map from $\G$ to $\{0,1\}$ and for $x \in \G$, $\boott{{d,\ell}}{\G}(x)=1$ if $x$ is occupied and $0$ otherwise. We will use the following basic facts: \begin{enumerate} \item[a)] The final configuration of the bootstrap process is a monotonic increasing function of the initial configuration. \item[b)] The updating procedure gives the same final configuration if applied to any configuration larger than the initial configuration and lower than the final one. \end{enumerate} In particular, b) implies that the updating order does not affect the final configuration. We say that a finite set $\G \subset \Z^d$ is {\it internally spanned} if $\boot{{d,\ell}}{{\G}}{x}=1$ for any $ x \in \G$. We focus our attention on the behavior of the following probability: \begin{equation*} R(L,p,\dl):= \P^d_p \tonda{ \forall x \in \LL{d}{L} \quad \boot{\dl}{\LL{d}{L}}{x}=1 } \end{equation*} We call {\it $\G$-clusters} the maximal connected sets of occupied sites in $X^{{d,\ell} }_{{\G}}$. Notice that all clusters are internally spanned. %[migliorare] We say that {\it $x$ is connected to $y$ in $\G$} if there exists a $\G$-cluster $\cC$ such that $\{x,y\} \subset \cC \subset \G$; we denote this event by $\left\{ \EEa{x}{y}{\G} \right\}$. %We consider the event of two simultaneous %"compatible" connections: %we denote by %$\left\{ \EE{x}{y}{d}{l} \right\} \compo %\left\{ \EE{w}{z}{d}{l} \right\}$ %the event %"there are two clusters %$\G_1$ and $\G_2$ %such that %$\{ \G_1 \cup \partial \G_1 \} \cap %\{ \G_2 \cup \partial \G_2 \} = \emptyset$, %$\{x,y\} \subset \G_1 \subset \LL{d}{l}$ %and %$\{w,z\} \subset \G_2 \subset \LL{d}{l}$" We will use the symbols $c$, $C$ and $\cc$ for positive constants (possibly depending on the parameters of the bootstrap percolation model). %and the symbol $\cc$ for arbitrarily large constants. \section{ \label{Section 3.} Main result. } \ The following theorem describes the threshold regime of finite volume bootstrap percolation for all values $3\leq \ell\leq d$. \begin{thm} For $2 < \ell \le d$, there exist $2$ constants $0<\costm(d,\ell) \leq \costp(d,\ell) < \infty$, independent of $p$, such that if \begin{equation} \label{eq:defl} L_{\pm} (\dl,p) := \eexp{(\ell-1)} \tonda{ \ca_{\pm} \ p^{\scriptstyle {-\esp }}}, \end{equation} then $$ \hbox{\rm a) } \P^{d,\ell}_{p} \tonda{\forall \; x \in \LL{d}{L} \quad X^{{d,\ell} }_{{\LL{d}{L}}}( {x}) = 1 } \to 1\ \ \ \text{ if } (p,L) \to (0,\infty ) \ \hbox{ with } L \ge L_+(\dl,p) $$ $$ \hbox{\rm b) } \P^{d,\ell}_{p} \tonda{\forall \; x \in \LL{d}{L}\quad X^{{d,\ell} }_{{\LL{d}{L}}} ({x}) = 1 } \to 0\ \ \ \text{ if } (p,L) \to (0,\infty ) \ \hbox{ with } L \le L_- (\dl,p) $$ \end{thm} {\bf Remark:} The case $2=\ell \le d$ is handled in \cite{[AL]}. The result in the case $\ell=d$ was a conjecture proposed in \cite{[ADE]}. The specific case $\ell=d=3$ was solved in \cite{[CeCi]}. \medskip It looks like the phenomenon hidden behind this behavior is linked with the notion of "critical droplet".%\nota{ dobbiamo citare van enter?} Indeed, the spanning probability has the same asymptotic behavior as the probability of finding in the volume $\LL{d}{L}$ a suitably large internally-spanned cluster. \medskip To prove this result, we use an inductive procedure. This is very natural for the estimate of the lower bound a). Indeed, the problem of the filling of a face of an hypercube once the hypercube is full is a bootstrap percolation problem with parameters $(d-1,\ell-1)$. By far less immediate is to see how to use induction in the proof of case b). We use there a natural generalization of the construction of Cerf and Cirillo, relating in this way a bootstrap percolation model with parameters $(d,\ell,p)$ with a bootstrap percolation model with parameters $(d-1,\ell-1,2p-p^2)$. \bigskip \section{Proof of case a)} \bigskip This is the easiest part of the proof. The argument is nothing new. In fact the idea of the argument is already present in \cite{[ADE]}. To estimate from below the spanning probability, we use iteratively Straley's argument and the renormalization procedure introduced in \cite{[AL]}. First, we use the renormalization scheme introduced in \cite{[AL]} and \cite{[S]} to prove that if $R(L^*,p,\dl ) \ge ({ e (2d-1)})^{-1}$ for some $L^*$, then $R(L,p,\dl) \ge 1 - C e^{-{L}/{L^*}}$ for all $L>L^*$. Assume (for simplicity's sake) that $L$ is an integer multiple of $L^*$. We tile $\LL{d}{L}$ with the translates of $\LL{d}{L^*}$. As initial condition for the bootstrap percolation process on the renormalized lattice we use the indicator functions of the events $$ \{ L^* x + \LL{d}{L^*}\text{ is internally spanned}\} \,,\quad x\in{\Bbb Z}^d\,.$$ It is clear that if the bootstrap process defined on the renormalized lattice spans the volume, so does the process on the original lattice. Hence, \begin{equation} \label{rinormalizzazione} R (L,p,\dl) \ge R \tonda{\frac L{L^*},R(L^*,p,\dl),\dl} \end{equation} If a box is not spanned then in the initial condition there must exist a cluster of empty sites that crosses the box. A standard site-percolation estimate, based on a Peierls type argument, gives \begin{equation} \label{pierls} 1-R (N,q,\dl) \le \sum_{l=N}^{\infty} {q}^{l} 2d (2d-1)^{l-1} = \frac{2d}{2d-1} \ \frac{ \tonda{(2d-1) q }^{N-1}} {1- (2d-1){q}} \end{equation} By \eqref{rinormalizzazione} and by \eqref{pierls} with $N=L/L^*$ and $q=R (L^*,p,\dl) \ge {(e (2d-1))}^{-1}$, we get \begin{equation} R (L,p,\dl) \ge 1- C {e}^{-{L}/{L^*}} . \label{eq:maggiorata} \end{equation} Next, we prove by induction the following property: there exists $ \costa_+ (\dl)>0$ such that if we set \begin{equation*} m_+ (\dl,p) := \defmpiu{d}{\ell} \end{equation*} then \begin{equation} \label{hpr1} \forall L>m_+(d,\ell,p)\qquad R(L,p,\dl) \ge 1- \exp{\tonda{- \frac{L}{L_+ (\dl,p)}}} \end{equation} %In the case $\ell = 1$, a single %occupied site in the initial condition is sufficient to span %the entire volume. %Hence, $\forall \; L \in \N \; , p \in [0,1]$, %\begin{equation} % \label{elle1} % R(L,p,\dl)= 1- \tonda{1- p}^{L^d} % \ge % 1- \exp \tonda{ -p L^d} %\end{equation} %For $L \ge L_+(d,1,p) = \costp(d,1) p^{-\frac{1}{d}}$, we get %$R(L,p,\dl) \ge 1- e^{\costp(d,1)} = ({e (2d -1)})^{-1}$ %for $\costp(d,1) = - \ln \tonda{1- ({e (2d -1)})^{-1}}$. For $\ell = 1$ and $\costp(d,1)=1$, \eqref{hpr1} is immediate, since a single occupied site in the initial condition is sufficient to span the entire volume. In the case $\ell = 2$, \eqref{hpr1} has been proven in \cite{[AL]} (see (1.5) therein). We end our induction proof by showing that if \eqref{hpr1} holds for $(d-1,\ell-1)$ then $R (L^*,p,\dl) \ge ({ e (2d-1)})^{-1}$ for $L^* \ge L_+(\dl,p)$. %Let $\cE_{\dl}$ be the event %$$ % \graffa{\exists \text{ in }\LL{d}{L} \text{ at time }0 % \text{ a translate of } \LL{d}{L_+(d-1,\ell-1,p) } % \text{ completely occupied}}. %$$ Let $m_+:=m_+ (\dl,p)$ (we drop the dependency of $m_+$ on $\dl,p$ to lighten the notation). In the case $2 < \ell \le d$, for $\costa_+ (\dl) > \costp (d-1,\ell-1)$ and sufficiently small $p$, we have that \begin{equation} m_+ \ge L_+^2(d-1,\ell-1,p)\,. \label{eq:l2} \end{equation} To estimate from below the probability that $\LL{d}{L}$ is spanned, we consider the event \smallskip 1) at time $0$ there exists in $\LL{d}{L}$ a box $x \; m_+ + \LL{d}{m_+} $ completely occupied and 2) for every $m_+ \le k \le L$ the $(d-1)$-dimensional bootstrap percolation models with parameters $(p,\ell -1)$ restricted to the faces of the boxes $x \; m_+ + \LL{d}{k}$ are internally spanned. %(see \cite{[S]} for more details on this construction). The idea %behind this construction is that once a box $x \; m_+ + \LL{d}{k}$ is occupied, the sites on a face of the box have an occupied neighbor in the box and therefore need only $\ell-1$ neighbors in the face to become occupied. This procedure can be iterated to fill the whole $\LL{d}{L}$.\footnote{We warn the reader that this argument is sligthly oversimplified since we are not considering the edges. We refer to \cite{[S]} for the full construction.} Thus, \begin{equation*} R(L,p,\dl) \ge \tonda{1- \tonda{1-p^{m_+^d}}^{\tonda{\frac {L}{m_+}}^d} } \prod_{k=m_+ +1}^{L} R(k,p,d-1,\ell-1)^{2d} \ge \end{equation*} \begin{equation*} \tonda{1-e^{{-p^{m_+^d}}{\tonda{\frac {L}{m_+}}^d}}} \exp \tonda{ - \sum_{k=m_++1}^{L} 2d \; e^{- \tonda{\frac{k}{\sqrt {m_+}}}}} \ge \end{equation*} \begin{equation} \label{tendea1} \tonda{1-e^{{-p^{m_+^d}}{\tonda{\frac {L}{m_+}}^d}}} \exp \tonda{ - 2d \sqrt {m_+} \; e^{- c \sqrt {m_+}} } % \frac{1}{1-e^{- m_+^{-1/2}}} } \end{equation} The factor $\sqrt {m_+}$ in the above formula \eqref{tendea1} comes from \eqref{eq:l2}. The last term in \eqref{tendea1} clearly tends to one for $M \to \infty$. By using \eqref{eq:defl} we see that also the first term goes to $1$ as $p \to 0$ for any $L \ge L_+(\dl,p)$: Indeed, for sufficiently small $p$, \begin{eqnarray} m_+p^{-\frac{m_+^d}{d}} \le p^{-c m_+^d}= & \exp \tonda{c \ln p^{-1} \tonda{ \exp \tonda{ d \eexp{(\ell-3)} \left( {\costa_+(d,\ell)}{p^{{-\frac{\scriptstyle 1}{\scriptstyle d-\ell+1}}}}\right)}}} \nonumber\\ \ & \le \eexp{(\ell-1)}\left( {\ca}{p^{{-\frac{\scriptstyle 1}{\scriptstyle d-\ell+1}}}}\right) \label{eq:spiega} \end{eqnarray} for sufficiently large $\costp (d,\ell)$, r.h.s. of \eqref{eq:spiega} tends to infinity slower than $L_+$ and for $L \ge L_+$, r.h.s. of \eqref{tendea1} can be bounded by $({ e (2d-1)})^{-1} $. We can then use the bound in \eqref{eq:maggiorata} and get \eqref{hpr1}. %Property \eqref{hpr1} implies that the spanning %probability can be bounded by the probability that %a completely full translate of the %hypercube $\LL{d }{ m_+ (\dl) } $ is found in the %initial configuration. %Hence, %\begin{equation} %\label{p1} %\P_p^{\dl} \tonda{ %\boot{d}{\ell}{x} %\equiv 1 \ %\forall \; x \in \LL{d}{L} } \ge %1 - \tonda{1- p^{ m_+^d (\dl) } }^{(\frac{L}{ m_+ (\dl)})^d} %\ge %\end{equation} %$$ %\ge %1-\exp \tonda{- \frac{L}{ L_+ (d-1,\ell-1,p)}}. %$$ %In the last inequality, we used that %$L_+(\dl,p) >> p^{-c m_+^d (\dl) }$ for any constant $c$. %Supposing that \eqref{hpr1} is valid for $(\dl)$, it %is easy to show that it also holds for $(d+1,\ell+1)$. %The key argument is that a %face of a completely full ($d+1$)-dimensional hypercube %with side-length $l$ is a %$d$-dimensional cube where all the sites have at least %one occupied neighbor in the last dimension. %Hence, %$$ %\P_p^{d+1,\ell+1} \tonda{ %\boot{d+1}{\ell+1}{x} %\equiv 1 \ %\forall \; x \in \LL{d+1}{L} | %\LL{d +1}{ m_+ (d+1,\ell+1) } \text{ full at time } t=0} %\ge %$$ %$$ %\ge %\prod_{k = \lceil (m_+ (d+1,\ell+1) -1)/2 \rceil }^{L} %\tonda{1 - \tonda{1-\P_p^{\dl} \tonda{ %\boot{d}{\ell}{x} %\equiv 1 \ %\forall \; x \in \LL{d}{ 2k+1} } %}} %\ge %$$ %$$ %\ge %\prod_{k= +1}^{L- \lceil m_+ (d+1,\ell+1) \rceil } %\tonda{1-\exp \tonda{- c \frac{ m_+ (d+1,\ell+1)+2k}{ L_+ (d-1,\ell-1,p)}}} %\ge %$$ %\begin{equation} %\label{hpr1b} %\ge %\exp \tonda{- c' L \exp \tonda{-\frac{m_+(\dl)}{L_+(d-1,\ell-1,p)}}}^d %\mioto 1. %\end{equation} %[mettere a posto i $k$] %The basis of the induction is the result of \cite{[AL]} %for the case $d \ge \ell =2$. %[Oppure il caso $\ell=1$] \bigskip \section{Proof of case b)} In order to prove part b) of the Theorem, we give a bound on the probability that two points are in the same cluster for the bootstrap percolation process in a box with diameter of the order of the critical droplet. Since we choose the box with a sufficiently small diameter, the final configuration is "sub-critical" and looks like subcritical site percolation. This is the content of our key estimate. Let us set \begin{equation} \label{defmmoins} \mm{d}{\ell}:= \defmmeno{d}{\ell}, \end{equation} where $\costa_- ({d,\ell})$ is a constant independent from $p$. \begin{lemma} \label{L1} Let $d \ge 1$ be fixed. For $\ell = 1$: \begin{equation} \forall m \in \N\qquad \P^d_p \left( \EE{x}{y}{d}{m}\right) = 1-(1-p)^{m^d} \label{hpr21} \end{equation} For $d \ge \ell = 2$: there exist $ \costa_- (d,2) > 0$, $C >0$ and $p(d,2)>0 $ such that: \smallskip $\forall p
0$, $ \costc(d,\ell) > 0$, $ p(d,\ell)>0 $ such that: \smallskip $\forall p
2$
to get
$$
R(L,p,\dl)
\le
L^{3d+1} p^{\cc \fracscript{L}{3}}.
$$
If $L \ge m_- (\dl,p) $,
we choose $\k= m_- (\dl,p) / 3$ and
we get
$$
R(L,p,\dl)
\le
L^{3d+1} p^{\cc {m_- (\dl,p)}/{3}}.
$$
From these inequalities, we see that
there exists $\costm (\dl) > 0$
such that if
$$
L \le
\eexp{(\ell-1)} \tonda{ \ca_{-} \ p^{\scriptstyle {-\esp }}},
$$
then $R(L,p,\dl)$ goes to $0$ in the limit
where $(L,p) \to (\infty,0)$.
\medskip
{\it Proof of Lemma 5.2.}
The result for $\ell=1$ is immediate, since
a single occupied site in the initial configuration
is sufficient to span the entire volume.
For the case $\ell=2$, we use a procedure
introduced by Aizenman and Lebowitz in \cite{[AL]}.
We consider an integer
\begin{equation*}
m \leq \mm{d}{2}=
\costa_-(d,2) \; p^{{-\frac{\scriptstyle 1}
{\scriptstyle d-1}}},
\end{equation*}
and we set
$q:=2p-p^2.$
Let $x \in \Z^d$ be a $d$-dimensional vector;
we denote by $\ux$ its first $d-1$ coordinates and
by $\bx$ the last one.
We write $x=\xd{\ux}{\bx}$.
By symmetry, we can suppose that
$\xd{\ux}{\bx}$ and $\xd{\uy}{\by}$
in $\Z^d$ are
such that $\by-\bx = \norma{\xd{\ux}{\bx} - \xd{\uy}{\by}}$,
namely that
the distance along the $d$-th direction is larger than
or equal to the distance in the other directions.
%$|\bx-\by| \ge \max_{1 \le d-1} |x_i - y_i|$.
We consider the slices
$$
\slice{i}:=\left\{ (\ux,\bx) \in \LL{d}{m} \ ;
\ \bx \in \{2i, 2i+1 \} \right\},\ i \in \Z.
$$
Suppose $\{\EE{x}{y}{d}{m}\}$ occurs.
Let $\cC$ be the $\LL{d}{m}$-cluster that contains
$x$ and $y$ and let $A$ and $B$ the first and the
last indices of the slices intersecting $\cC$.
It is immediate to see that in all the slices
$\slice{i}$ for
$i \in [A,B]$
there exists at least one occupied site $\xd{\ux',\bx'}$
such that $\norma{\ux - \ux'} \le \norma{x - y}$.
Let us set $l:=\norma{x-y}$.
By \eqref{hpr21},
\begin{equation}
\label{ceci1}
\P^d_p \left( \EE{x}{y}{d}{m}\right)
\leq
d \tonda{1-\tonda{
1-q}^{{
(2 l +1 )}^{d-1}}}^{ l /{2}},
% \leq
\end{equation}
where the factor $d$ comes from the possible
directions where $\norma{x-y}$ is realized.
By using the fact that
$\ln t \le t - 1$,
we bound the r.h.s. of \eqref{ceci1} by
%$$
% \exp \tonda{\frac{\norma{x-y}}{2}
% \ln (1- \exp({(2\norma{x-y}+1)^{d-1} }\ln (1-q)))}
% \leq
%$$
\begin{equation*}
%\label{ceci}
% \leq
d \tonda{ - \ln (1-q)^{{(2 l +1)}^{d-1}}}^\frac{l}{2};
\end{equation*}
for small $q$, $ \ln (1-q) \ge - 2 q $ and hence we get
bound \eqref{hpr22}.
%The factor $d$ in \eqref{ceci} comes from the number
%of possible directions where $\norma{x-y}$ is realized.
For $\ell \ge 3$, we use an induction on the dimension
$d$ and on the parameter $\ell$.
%:for $\ell \ge 4$, we assume the thesis is true for
%$d-1$, $\ell -1$ and prove it for $d$, $\ell$;
%then
%we show that in the case $\ell=3$ \eqref{hpr2}
%follows from \eqref{hpr22}.
Following \cite{[CeCi]}, we define an auxiliary map $\ZZ{d}{\LL{d}{m}}{x}$ on
$\Z^d$
in the following way.
In every slice $\slice{i}$,
we increase the initial configuration
by occupying a site $(\ux,2i)$ (resp $(\ux,2i+1)$)
if the corresponding site
$(\ux,2i+1)$ (resp. $(\ux,2i)$)
belonging to the other hyper-plane in the
same slice is occupied.
We build a configuration $\YY{d}{\LL{d}{m}}{(\ux,\bx)}$
by updating on each slice this initial configuration according
to the bootstrap percolation process
in $\slice{i}$ with the neighboring slices occupied;
more precisely, we occupy
$\LL{d}{m}\setminus \slice{i}$,
we run $\boott{d,\ell}{\LL{d}{m}}$
under this initial condition
and we define
$\YY{d}{\LL{d}{m}}{\ux,2i)}=\YY{d}{\LL{d}{m}}{\ux,2i+1)}$
as the restriction to $\slice{i}$ of this
bootstrap percolation process.
The monotonicity properties of bootstrap percolation
with respect to the initial configuration imply that
$\YY{d}{\LL{d}{m}}{(\ux,\bx)} \ge
\boot{{d,\ell}}{\LL{d}{m}}{(\ux,\bx)}$.
The interesting point is that
%$\YY{d}{\LL{d}{m}}{(\ux,2i)}=\YY{d}{\LL{d}{m}}{(\ux,2i+1)}$
%and that
the set
$\{ \ux \in \LL{d-1}{m} \ ; \
\YY{d}{\LL{d}{m}}{(\ux,2i)}=1 \}$
is equal to the
($d-1$)-dimensional bootstrap of the ($d-1$)-dimensional
configuration where the site
$\ux\in \LL{d}{m}$ is occupied if either
$\xd{\ux}{2i}$ or $\xd{\ux}{2i+1}$ was initially occupied.
%$$
%\{ \ux \in \LL{d-1}{m} \ ; \
%\iniz{d}{\LL{d}{m}}{(\ux,2i)}=1 \text{ or }
%\iniz{d}{\LL{d}{m}}{(\ux,2i+1)}=1\};
%$$
Thus,
$\YY{d}{\LL{d}{m}}{(\ux,\bx)}$ is a stack of
($d-1$)-dimensional bootstraps with parameters $\ell-1$
and $q=2p-p^2$.
We set
\begin{equation*}
n:=\defn{d}{\ell},
\end{equation*}
for a suitable constant $\costb(d,\ell)$
which will be chosen later on (see before \eqref{l=3}).
%\nota{In
%effetti, usiamo $\costb(\dl)$ solo nel caso $\ell=3$, altrimenti
%nella \eqref{somma} lo
%maggioriamo con $\infty$!}
We finally define our process as
$\ZZ{d}{\LL{d}{m}}{(\ux,\bx)}:=\YY{d}{\LL{d}{m}}{(\ux,\bx)}$
if the slice containing $(\ux,\bx)$ does not contain any
cluster larger than $n$;
otherwise, we set
$\ZZ{d}{\LL{d}{m}}{(\ux,\bx)}:=1$
for all the sites
$(\ux,\bx)$ in the slice.
For $\G \in \LL{d}{m}$,
we denote by
$
\big\{ \EEEa{\ux}{\uy}{\G} \big\}
$
the event
$$\exists\,\cC
%there is a connected set
\subset \slice{i},\quad
\cC\text{ is connected},\quad
\{(\ux,2i),(\uy,2i)\} \subset \cC,\quad
\forall \; (\uy',\by') \in \cC\quad
\ZZ{d}{\LL{d}{m}}{(\uy,\by)}=1\,.$$
By construction, we get for all $i \in [A,B]$
\begin{equation*}
\P^d_p \left(\big\{ \EEE{\ux}{\uy}{i}\big\}
\cap
\big\{
\; \forall \; y \in \slice{i} \quad
\ZZ{d}{\LL{d}{m}}{y}=1\big\}^c
\right)
\le
\P^{d-1}_q \left(
% \EE{\ux}{\uy}{d-1}{m}
\ux \longconx{d-1,\ell-1} \uy \ \text{\rm in } \LL{d-1}{m}
\right).
%\label{dmu}
\end{equation*}
Let $\{I_k\}_{k \leq U}$ be the ordered set of indices,
between $A$ and $B$
of the slices that are completely full in
$Z^{d}$ i.e.
$
\big\{k ; \ZZ{d}{\LL{d}{m}}{(\ux,2I_k)}=1
\ \forall \; \ux \in \LL{d-1}{m}\big\}
$.
We set $I_0:=A$, $I_{U}:=B$
We decompose our event $\big\{\EE{x}{y}{d}{m}\big\}$
according to the possible values of
$A$, $B$, $U$ and $\{I_k\}_{k \leq U}$.
We have
$$
\P^d_p \left( \EE{x}{y}{d}{m} \right) \leq
d
\sum_{a=1}^m
\sum_{b=\by-\bx+a}^{m} \
\sum_{u=0}^{b-a}
\sum_{i_1<\cdots