95-428 Knill O.
Determinants of random Schroedinger operators arrizing from lattice gauge fields (68K, LaTeX) Sep 21, 95
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Abstract. For a class of bounded random selfadjoint operators $(Lu)(n)=\sum_{i=1}^d A_i(n) u(n+e_i)+A_i^*(n-e_i) u(n-e_i)$, determined by a discrete abelian or nonabelian $U(N)$ lattice gauge field $n \mapsto (A_1(n), \dots, A_d(n))$ on $\ZZ^d$, the potential theoretic logarithmic energy $I(L)=-\int \int \log |E-E'| \; dk(E) \; dk(E')$ of the density of states $dk$ of $L$ is finite and satisfies $I(L)=-\log|\det(L^{(2)}|$, where $L^{(2)}$ is the two particle Hamiltonian of $L$. For the $n$-particle Hamiltonians $L^{(n)}$ defined on the $n$ particle subspace of the Fock space, we show the existence of ergodic or gauge invariant minimizers of the height functionals $I_n(L)=-\log |\det(L^{(n)})|$. We prove $I_n(L) \in [-\log(\sqrt{2nd}),0]$ and $I_n(L) \sim {\rm EulerGamma}/2-\log(\sqrt{dn}) + o(1)$ for $n \rightarrow \infty$. A random walk expansion for $I_{n,\beta}(L)=-\log |\det(L^{(n)}-\beta)|$ identifies $\det(1-z L^{(n)})$ as a dynamical zeta function.

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