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\title{Determinants of random Schr\"odinger
operators arrizing from lattice gauge fields}
\author{Oliver Knill \thanks{Division of Physics, Mathematics and Astronomy,
California Institute of Technology, 253-37,
Pasadena, CA, 91125 USA. } }
\date{September 21, 1995}
\begin{document}
\bibliographystyle{plain}
\maketitle
\begin{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.
\end{abstract}
{\bf Keywords:} Random lattice gauge fields, potential theory of the spectrum,
Discrete random Schr\"odinger operators.
\pagestyle{myheadings}
\thispagestyle{plain}
\section{Introduction}
A class of random Schr\"odinger operators can
be obtained from abelian or nonabelian discrete random fields on $\ZZ^d$. These
operators have finite determinants.
A natural problem is to minimize $-\log|\det L|$
among this class of operators, to determine the set of extrema as well
to study their properties.
This variational problem is mathematically
similar to the entropy problem in statistical mechanics
and has relations with lattice gauge fields, random matrix theory,
random Schr\"odinger operators and potential theory in the complex plane.\\
Our guide is the variational problem
in statistical mechanics to maximize the
entropy on the set of invariant measures of
an expansive topological dynamical system $(X,T)$
with time $\ZZ^d$. Because this functional is upper semi-continuous,
it takes its maximal value,
the topological entropy. (See for example \cite{Ruelle}). \\
Every discrete gauge field $x$ with the structure group $U(N)$
in $d$ dimensions defines an operator $L$ on $\Hcal=l^2(\ZZ^d,\CC^N)$ by
$(L_xu)(n)=\sum_{i=1}^d x_i(n) u(n+e_i)+x_i^*(n-e_i) u(n-e_i)$.
Given a measure $\mu$ on $X$ and a random variable $x \mapsto L_x$ taking values
in the space of these operators, there exists a density of states $dk_{\mu}$.
In lattice gauge field theory, one usually considers
the Wilson Hamiltonian $n \mapsto H_x(n)=(L_x^4)_{nn}$.
The expectation value $=\tr_{\mu}(L^4)$ has as a continuous function both
minima and maxima. We consider here the
Hamiltonian $n \mapsto -\log|L_x|_{nn}$ which is not
bounded, but which has, as we will see,
a finite expectation value $-\tr_{\mu} \log|L|=-\log |\det_{\mu}(L)|$.
A "van der Monde determinant" of an operator $L$ is
obtained by applying the functional calculus for $L$ to the "characteristic
function" $\beta \mapsto \det_{\mu}(L-\beta)$ and taking again the determinant
$\Delta_{\mu}(L)=\det_{\mu}(\det_{\mu}(L-L))$.
While $\Delta_{\mu}(L)$ is not
defined for many operators and not interesting for matrices, where
trivially $\Delta_{\mu}(L)=0$, the value
$\Delta_{\mu}(L)$ is nonvanishing and finite for random operators obtained
from lattice gauge fields and
$I_2(\mu)=-\log \Delta_{\mu}(L)$ coincides
with the potential theoretical energy of the density of states.
$I_2$ is lower semicontinuous
as a functional on the set $M(X)$ of shift-invariant
measures on the set $X$ of all gauge fields.
The minimizers of this energy exist therefore by a similar reason as
equilibrium measures exist for the entropy functional. \\
A variational problem for Schr\"odinger operators $L$
arrizing in discrete Hamiltonian systems $T: \TT^{2n} \rightarrow \TT^{2n}$
is closely related to the entropy maximizing
problem in ergodic theory because the maximal value of
$\log|\det_{\mu}(L)|$ is the topological entropy of the map,
if the maximizer $\mu$ is
absolutely continuous with respect to the volume measure on $\TT^{2n}$
and the random operator $L$ is the Hessian of a discrete action functional
at a critical point. \\
A finite dimensional version of the
problem to minimize $I(\mu)$ is the following: consider
a finite discrete torus which is the Cayley graph of the group $\ZZ_p^d$ with
generators $e_1, \dots ,e_p$. If we attach to each bond $(n,e+e_i)$
an element $x_i(n) \in U(N)$,
this defines a finite gauge field
$x(n)=(x_1(n), \dots, x_d(n)), \; n \in \ZZ_p^d$.
Consider the finite dimensional operator $(L_x u)(n)
= \sum_{i=1}^d x_i(n) u(n+e_i) + x_i^*(n-e_i) u(n-e_i)$
on the finite dimensional space $l^2(\ZZ_p^d,\CC^N)$.
The flux-phase
problem is to minimize $-\tr(|L|)=-\sum_i |\lambda_i|$ and a related question
is to minimize $-|\det(L)|=-\prod_{i} |\lambda_i|$, where $\lambda_i$ are
the eigenvalues of $L$
(see \cite{Lie92,LiLo93,Lie94}).
The finite dimensional version of the problem to minimize the electrostatic
energy of the density of states is to minimize
$I(L)= - \log \prod_{iFrom the probabilistic point of view, the Poisson
distribution is the most natural
choice. We denote this trace by
$$ {\tr}_{\lambda}(\overline{L})
= \sum_{n} e^{-\lambda} \frac{\lambda^n}{n!} \tr(L^n) \; .$$
The bilinear Hamiltonian $\overline{L}$
has a density of states which
we denote by $\overline{dk}_{\lambda}$.
A parameter is the
expected number of particles $\tr_{\lambda}(\overline{N})=\lambda$, where
$\overline{N}$ is the {\it number operator} on the Fock space which takes
the constant value $n$ on $\Hcal^{(n)}$. \\
The tensor product of ergodic operators is ergodic:
\begin{propo}
If $K_i, \; i=1, \dots n$ are ergodic operators
defined over $\ZZ^d$-dynamical system $(\Omega_i,T_{ij},m_i)$, where
$T_{ij}: \Omega_i \rightarrow \Omega_i$, $j=1, \dots, d$. Then
$K_1 \otimes \dots \otimes K_n$ is a random operator
over the ergodic $\ZZ^{nd}$ dynamical system
$(\prod_{i=1}^d \Omega_i,T_{ij},m_1 \otimes \cdots \otimes m_n)$.
\end{propo}
\begin{proof}
Given a bounded measurable
function $f: \prod_{i=1}^d \Omega_i \rightarrow \RR$,
which is invariant under all transformations $T_{ij}$. Since by
the ergodicity of $\{T_{ij}\}_{j=1\dots d}$ for
every $j=1, \dots, n$, the function
$x_j \mapsto f(x_1, \dots, x_j, \dots, x_n)$ is constant for $m_j$-
almost all points in $\Omega_j$, also $f$ is constant almost everywhere.
\end{proof}
Remarks. \\
1) It follows that if $L$ is an ergodic operator,
then $L^{(n)}$ is an ergodic operator and the spectrum
of $x \mapsto L_x^{(n)}$ is almost everywhere constant. \\
2) For $d=1$, multi particle operators $L^{(n)}$
allow isospectral Toda deformations.
This can be seen directly or using the criterium in \cite{kni94}.
\section{A sequence of variational problems}
A random Schr\"odinger operator has
the determinant $|\det_{\mu} L|=e^{\tr_{\mu}(\log|L|)}
=e^{\int \log|E| \; dk_{\mu}(E)}$.
Borrowing terminology from differential geometry, we call
$-\log| \det L|= -\tr(\log|L|)$ a {\it height}.
Also the operator $\overline{L}$ has a determinant with corresponding height
$\overline{I}=-\log|\det_{\lambda} \overline{L}|
=-\tr_{\lambda}(\log |\overline{L}|)$.
We consider the problem to minimize
$I_n(\mu)=-\log |\det_{\mu}(L^{(n)}|)$ or
$\overline{I}$ on the set $M(X)$ of all shift
invariant probability measures $\mu$ on $X$. \\
Any random operator is an operator-valued random variable $x \mapsto L_x
\in \Bcal(l^2(\ZZ^d,\CC^N))$.
As for real-valued random variables, two
random operators are called {\it independent}, if the $\sigma$-algebras,
they generate, are independent.
For example, if $L,K$ are random operators, then $L \otimes 1$ and
$1 \otimes K$ are independent.
\begin{lemma}
\label{independent}
If $L_1,L_2, \dots, L_n$ are independent random operators,
then the density of states of
the sum $L=\sum_i L_i$ is the convolution $dk_1 \otimes \dots \otimes dk_n$
of the density of states $dk_i$ of the individual operators $L_i$.
\end{lemma}
\begin{proof}
The operators $L_i$ are independent operator-valued random variables.
The real-valued random variables $l_i: \omega \mapsto {\rm tr}(L_i)$ are
independent too
and their law is just the density of states $dk_i$. Since $l=\sum_{i=1}^d l_i$
is the trace of $L$,
the claim follows from the fact that
the law of a sum of independent random variables is the convolution
of the laws of the summands.
\end{proof}
Remark. It follows
that the set of measures occurring as the density of states of
random operators attached to gauge fields is closed under convolutions.
\begin{coro}
The height of $L^{(n)}$ satisfies
$$ I_n(\mu)= -\log |\det_{\mu}(L^{(n)})| = -\int_{\RR} \log |E| dk_{\mu}^{(n)}(E)
= -\int_{\RR^d} \log|E_1 + \dots + E_n| \;
dk_{\mu}(E_1) \dots dk_{\mu}(E_n) \; . $$
\end{coro}
\begin{proof}
By definition of the convolution of measures on $\RR$, one has
$\int f(E) \; dk \otimes dk(E) = \int \int f(E+E') \; dk(E) \; dk(E')$
and more generally
$$ \int_{\RR} f(E) \; dk^{(n)}(E) = \int_{\RR^d} f(E_1+E_2 + \cdots + E_d)
\; dk(E_1) \; \dots \; dk(E_d) \; . $$
\end{proof}
Especially:
\begin{coro}
The potential theoretical
energy of the density of states $dk$ of $L$ is the height of
$L^{(2)}$ attached to $L$:
$$ I(\mu)=-\log |\det(L_{\mu}^{(2)})| \; . $$
\end{coro}
\begin{proof}
Because by Lemma~(\ref{symmetric}), $dk$ is symmetric,
\begin{eqnarray*}
\log |\det_{\mu} L^{(2)}|
&=&
\int_{\RR} \log|E| \; dk_{\mu}^{(2)}
= \int_{\RR^2} \log|E+E'| \; dk_{\mu}(E) \; dk_{\mu}(E') \\
&=& \int_{\RR^2} \log|E-E'| \; dk_{\mu}(E) \; dk_{\mu}(E')
= - I(\mu) \; .
\end{eqnarray*}
\end{proof}
Remarks. \\
1) Because the mean of $dk$ is $\tr(L^{(n)})=0$
and the variance is $\tr((L^{(n)})^2)=2dn$,
the central limit theorem implies
that the density of states of the operator $L^{(n)}/\sqrt{2d n}$ converges to
the normalized Gauss measure on $\RR$. If $\sum_n p_n=1$, then $p_n \rightarrow 0$
faster than $1/\sqrt{2dn}$ so that the density of states of $p_n L^{(n)}$ converges
to the Dirac measure on $0$ which implies that $p_n L^{(n)}$ converges to
zero in norm. \\
2) The Fourier transform of $dk$, the characteristic function
$s \mapsto \hat{dk}(s)=\int e^{ist} \; dk(t) = {\tr}(e^{isL})$
contains all the information of the density of states. As in probability theory,
the knowledge of the Fourier transfor $\hat{dk}$
is useful for computations of $dk^{(n)}$ which has the Fourier transform
$(\hat{dk})^n(s)$.
\begin{propo}
\label{4.4}
Denote by $\gamma$ the Euler constant. Given any operator $L$, we have
$$ I_n(L) - \gamma/2+\log(\sqrt{dn}) \rightarrow 0, \;
n \rightarrow \infty \; . $$
\end{propo}
\begin{proof}
The density of states of $L^{(n)}$ is the law of the sum of $n$
independent random variables with law $dk$, zero expectation and
variance $\tr(L^2)=2d$.
For any real number, the density of states of $r L$ is $dk(rE)$ if
$dk(E)$ is the density of states of $L$. \\
The central limit theorem implies that the density of
states of $L^{(n)}/\sqrt{2dn}$ converges weakly to the Gauss measure
$(2 \pi)^{-1/2} e^{-x^2/2} \; dx$. Therefore, for ${\rm Im}(E) \neq 0$,
and $n \rightarrow \infty$
$$ G_n(E)=-\log |\det(L^{(n)}/\sqrt{2dn} + E)| \rightarrow
-(2 \pi)^{-1/2} \int_{-\infty}^{\infty}
\log|x+E| \cdot e^{-x^2/2} \; dx = F(E) \; . $$
By the dominated convergence theorem, the smoothed functions
$(\chi_{\epsilon} \star F)(0)$ converge for
$\epsilon \rightarrow 0$ to
$$ F(0)=(2 \pi)^{-1/2} \int_{-\infty}^{\infty}
\log|x| \cdot e^{-x^2/2} \; dx $$
which has the value $(\gamma+\log(2))/2$. Also
$(\chi_{\epsilon} \star G_n)(E)$ converges with fixed $n$
for $\epsilon \rightarrow 0$ to $I_n + \log(\sqrt{2dn})$.
\end{proof}
Remarks. \\
1) The asymptotic formula in Proposition~(\ref{4.4}) holds for any random
operator $L$ which satisfies $\tr(L)=0$ and $\tr(L^2)=2d$. \\
2) A reformulation of the law of iterated logarithm in the present context
is that the accumulation points of
$\tr(L^{(n)})/\sqrt{2nd \log \log(n)}$ is the set $[-1,1]$. \\
3) The operator $L^{(n)}$ has its spectrum in $[-2nd,2nd]$ and
for the equilibrium measures $dk_{[-2nd,2nd]}$ on this interval
we get the minimal energy $-\log(nd)$ which is smaller than
$-\log(\sqrt{dn})$. It follows that for large enough $n$, the minimizers
of $I_n$ do not achieve the potential theoretical minimum. We see
below that this is true unless $dn=1$. \\
4) The integral
$$ \lim_{n \rightarrow \infty} \log |\det(L^{(n)}/\sqrt{2dn}-\beta)|
= (2 \pi)^{-1/2} \int_{-\infty}^{\infty}
\log|x-\beta| \cdot e^{-x^2/2} \; dx $$
seems to be difficult to evaluate in closed form, but
it can be computed numerically. \\
5) Let $\overline{N}$
is the number operator on the Fock space. The
determinant $\det_{\lambda}(\overline{L} \; (2d \overline{N})^{-1/2})$ can
be computed in the
limit of an infinite expectation value of particles as
$$ \lim_{\lambda \rightarrow \infty}
\det_{\lambda}(\overline{L} \; (d \overline{N})^{-1/2})
= e^{\gamma/2} = 1.33457 \dots \; . $$
\section{Existence of minimizers}
\begin{propo}
The functionals $I_n$ and $\overline{I}$ take their minima
on the compact metric space $M(X)$ of shift invariant measures $\mu$ on $X$.
\end{propo}
\begin{proof}
The map $\mu \mapsto dk$ is continuous because for all $f \in C(\RR)$,
the map $\mu \mapsto \tr_{\mu}(f(L))= \int f(L)_{00} \; d\mu$
is continuous. Also $\mu \mapsto dk_{\mu}^{(n)}=\int f(L^{(n)})_{00} \; d\mu$
is continuous. \\
The map $E \mapsto \lambda_{n,\mu}(E)=\int \log|E-E'| \; dk_{\mu}^{(n)}(E')$
is subharmonic and bounded and therefore
the pointwise infimum of in $E$ smooth functions
$\lambda_{n,\mu,\epsilon}(E)
= (\chi_{\epsilon} \star I_{n,\mu}(\mu))(E)$. These functions
are continuous in $\mu$ for $\epsilon \neq 0$ because they are integrals
of bounded functions which have this property almost everywhere. \\
Therefore, $\mu \mapsto I_n(\mu)$ is lower semicontinuous as a pointwise limit of
continuous functions on
the compact metric space $M(X)$ and such a function has a minimum. \\
The same argument holds for the density of states $dk^{(n)}$ or $\overline{dk}
= \sum_{n=0}^{\infty} p_n dk^{(n)}$.
\end{proof}
\begin{coro}
There exist ergodic minimizers of $I_n,I,\overline{I}$.
\end{coro}
\begin{proof}
The maps
$\mu \mapsto -\log|\det_{\mu} L^{(n)}|$ $\in [-\log(nd),0]$ and
$\mu \mapsto -\log|\det_{\mu} \overline{L}|$ $\in (-\infty,0]$ are
linear in $\mu$ if we extend it to all signed measures. \\
Fix $n$. Given a probability measure $\mu$, on which $I_n$ is minimal.
We can write it as
an integral over ergodic probability measures and the minimality assures that
$I_n$ takes the minimal value on almost all of the ergodic measures
in the ergodic Choquet decomposition of $\mu$.
\end{proof}
Remarks. \\
1) If a nonergodic minimum exists, it is not unique because it can then
be written as an integral over ergodic measures, on which the functionals take
the same minimal value. \\
2) One could try to find the minimum of $I_n$ by determining
minimizers $\mu_{k}$ on the set of $k$-periodic
measures which have the support on $k$-periodic configurations.
But there is no reason why an
accumulation point of $\mu_{k}$ should be a minimizer for $I_n$. (For the
entropy functional $\mu \mapsto h_{\mu}(T)$,
there exists a dense set of periodic measures, which have
zero entropy but the maximum is positive.)
\section{Lower bounds on $I_n$}
Potential theory gave the lower bound $I=I_2 \geq -\log(d)$ which would
be achieved if the density of states of $L^{(n)}$ was the equilibrium
measure. We saw also from the central limit theorem that
$I_n \geq -\log(\sqrt{dn})$ for large enough $n$.
Using the Jensen inequality for random operators, one can say more: \\
\begin{lemma}
\label{Jensen}
For all $k \in \NN \setminus \{0\}$,
$$ I_n=- \log |\det(L^{(n)})| \geq - (1/k) \log \tr(|L^{(n)}|^k) \; . $$
\end{lemma}
\begin{proof}
(i) Jensen's inequality for the probability space $(\RR,dk)$
gives for any random operator $K$ and any convex bounded function $f$
$$ \tr(f(K)) = \int f(E) \; dk(E) \geq f(\int E \; dk(E)) = f(\tr(K)) \; . $$
(ii) In order to apply (i) with $K=|L^{(n)}|^{k}$
and the convex unbounded function
$f(x)=-\log|x|$, we approximate $f$ from above by bounded continuous convex
functions $f_l$. The dominated convergence theorem and the finiteness of
$-\log |\det_{\mu}(L)|=-\int f(L)_{00} \; d\mu$ assures that
$$ \int f_l(|L|^k)_{00} \; d\mu \rightarrow \int f(|L|^k) \; d\mu
= - k \cdot \log |\det_{\mu}(L)| \; . $$
>From (i), we know
$$ \int f_l(|L|^k)_{00} \; d\mu \geq f_l( \int (|L|^k)_{00} \;d\mu )
\rightarrow f( \int (|L|^k)_{00} \;d\mu ) = - \log \tr(|L|^k) \; . $$
\end{proof}
Remarks. \\
1) Applying this lemma in the case $k=1$ shows
that minimization of $-\log |\det(L^{(n)})|$
tells something about the minimum of $- \log \tr(|L^{(n)}|)$ and so about
the problem to minimize $- \tr|L^{(n)}|$. It follows
also that $\tr|L^{(n)}| \geq 1$. \\
2) Jensen's inequality has a generalisation in form of the
Peierl's-Bogoliubov inequality
$$ \tr(f(K) e^L)/\tr(e^L) \geq f( \tr(K e^L)/\tr(e^L)) $$
in finite dimensions (see \cite{Simon83}).
Using Avron-Simon's lemma \cite{Cycon}
this can easily be generalized to the random case.
\begin{coro}
$I_n(\mu) \geq - \log \sqrt{2nd}$ for all $n$ and all $\mu \in M(X)$.
\end{coro}
\begin{proof}
Apply b) in the case $k=2$ and use that $\tr((L^{(n)})^2)=2nd$, which is
the number of paths of length $2$ in $\ZZ^{nd}$ starting at the origin.
\end{proof}
Remarks. \\
1) It follows that $I=I_2 \geq - \log \sqrt{4d}$ which is better than
$I \geq - \log(d)$ given by the bound from the
equilibrium measure. Because, an operator with the equilibrium measure
on $[-2d,2d]$ satisfies
$\tr(L^2)=\int_{-2d}^{2d} \pi^{-1}x^2/\sqrt{4d^2-x^2} \; dx= 2d^2$
and in our case always $\tr(L^2)=2d$, the density of states
$dk(L)$ is for $d>1$ never
the equilibrium measure on $[-2d,2d]$. On the other hand, for
$d=1$, the density of states is always the equilibrium measure on $[-2,2]$. \\
2) The Green function $g(E)$ of the spectrum $\sigma(L)$ is related to
the electrostatic conductor potential $\lambda(E)$ by
$g(z)=-u(z)+\log(\gamma)$, where $\gamma$ is the capacity of the spectrum
(see \cite{Tsuji}). This shows that a random operator
has the equilibrium measure on the spectrum as the density of states
if and only if the function $\lambda(E)$ is constant on the spectrum
(it takes then the constant value $\log(\gamma)=-I(L)$ on the spectrum). \\
3) The tails of the density of states can be estimated using knowledge of
the moments $\tr(L^n)$. Let $Y$ be a random variable with law $P=dk$.
Chebychev-Markov's inequality gives for $c>0$
$$ 2 \int_{-\infty}^{-c} \; dk
= P[|Y| \geq c] \leq E[|Y|^n]/c^n = \tr(|L|^n)/c^n \; .$$
\section{Gauge invariant minimizers}
The compact topological group
$G=X=U(N)^{\ZZ^d}$ acts on $X$ by gauge transformations defined by
$$ x \mapsto (g(T_1) x_1 g^{-1}, g(T_2) x_2 g^{-1}, \dots, g(T_d) x_d g^{-1})\;,$$
where the transformations $T_i$ are the shifts on $X$. If a
measure $\mu$ is shift invariant, then, for
$g \in G$, the measure $g^* \mu$ needs no more
to be shift invariant in general. We still assign to $g^* \mu$
the same value $I(\mu)$ because gauge transformation induces a unitary
conjugation $L_x \mapsto g L_x g^{-1}$ for each operator $L_x$ on $\Hcal$.
\begin{propo}
There exists a minimum of $I$, which is both gauge and shift invariant.
The same holds for the functionals $I_n,\overline{I}$.
\end{propo}
\begin{proof}
The existence is shown in two steps.
We first construct a gauge invariant measure $\mu_0$ and construct from
it later a measure which is also shift invariant. \\
The Haar measure $\nu$ on $G$
defines a Borel measure $\rho=\phi^* \nu$
on the compact set $G^* \mu \subset M(X)$
by push-forward of the measurable map $\phi: g \mapsto g^* \mu$.
The functional $I$ takes the same constant value on the compact convex hull
of $G^* \mu$. By Choquet theory (see \cite{Phelps}),
there exists a unique measure $\mu_0 = \int_{G^* \mu} \mu' \; d\rho(\mu')$
on the convex hull of $G^* \mu$.
This measure is gauge invariant. We find therefore for each measure
$\mu$ a natural gauge invariant measure $\mu_0$ such that $I(\mu)=I(\mu_0)$.
If $\mu$ is a minimizer, then also the gauge invariant measure $\mu_0$
is a minimizer. \\
Given a gauge invariant measure $\mu_0$, then also $T_i^* \mu_0$ is gauge
invariant and $I_n(\mu_0)=I_n(T_i^* \mu_0)$.
With the van Hove sequence $\Lambda^n_0=[-n,n]^d \subset \ZZ^d$, a
shift and gauge invariant measure can be constructed by taking an accumulation
point of
$$ |\Lambda_0^n|^{-1} \sum_{k \in \Lambda_0^n} (T^k)^* \mu_0 \; . $$
Since $I_n$ is lower semicontinuous, the value of $I_n$ on any
of these accumulation points is equal to the minimal value $I(\mu_0)$.
\end{proof}
\begin{propo}
\label{moduli}
There exists a convex compact metric space $Y$ of
measures on $X$ which represents the gauge equivalence classes
of measures. Different points in $Y$ represent different
equivalence classes.
\end{propo}
\begin{proof}
Since the map $\psi: \mu \mapsto \mu_0$
described in the last proof is continuous, the image
$Y$ of $\psi$ is compact and represents all gauge and shift invariant measures.
Any convex combination of gauge invariant measures is
again gauge invariant. If $\mu$ is gauge invariant,
then $\mu_0=\mu$ so that $Y$ consists of all gauge invariant measures.
\end{proof}
Remarks. \\
1) It is not excluded that different gauge equivalence classes are
represented by the same point in $Y$, but then, $I,I_n$ and $\overline{I}$
take the same value on all of these equivalence classes. \\
2) The gauge invariant measure $\psi(\mu) \in Y$ belonging to $\mu$
might not be ergodic. We do not know
whether a gauge invariant ergodic minimum exists. \\
3) The Haar measure $\mu_{Haar}$ on $X$ is an
ergodic shift and gauge invariant measure on $X$.
It is a candidate for the minimum. \\
4) For $d=2$, the energy of the equilibrium measure on $[2d,2d]$
is $-\log(2)=-0.69315$. We
got a numerical value $I_2(L_{free,d=2})=-0.40533$. \\
5) The computation of $I_n$ for the free operator $L_{free,d}$
can be done as follows. The density of states of $L_{free,d}^{(n)}$
is $dk_{free}^{(d n)}$, where $dk(free)$ has the arcsin distribution on
$[-2,2]$. Therefore, $dk_{free}^{(d n)}$ is the law of $\sum_{i=1}^{nd} Y_i$
of independent random variables $Y_i$ with distribution function
$F(t)=1/2 + 1/\pi \arcsin(t/2)$. In other words, we can write
the functional $I_n$ for the free operator as the expectation
$$ I_n(L^{(n)}_{free,d}) = -\log |\det L^{(n)}_{free,d}| =
- {\rm E}[ \log | \sum_{i=1}^{nd} Y_i |] \; . $$
For numerical purposes, it is convenient
to take independent random variables $Z_i$
with uniform distribution on $[0,1]$ and to compute
$$ J_{nd}:=I_n(L_{free,d})
= - {\rm E}[ \log(| \sum_{i=1}^{nd} 2 \sin(\pi(Z_i-\frac{1}{2})) |) ] \; . $$
This can be computed by a Monte Carlo method. We know that
$J_{nd}=I_n(L_{free,d})=0$ for $nd=1,2$. Our numerical experiments gave
$J_3 \sim -0.32, J_4 \sim -0.42, J_5 \sim -0.54, J_6 \sim -0.65$ but these
data are not reliable. It would
be good to have some negative upper bounds. \\
6) The functionals $\tr(L^k)$ take the maximal value for $L=L_{free,d}$
for every $k$. It is not sure, whether the functionals
$I_n$ have a maximum. If yes, then $L_{free,d}$ is a candidate.
\section{Operators with a given curvature}
Given a random operator $L$ defined by a measure $\mu \in M(X)$.
The {\it field} of $L$ is defined as
$$ dx_{ij}(n)=F_{ij}(n) = x_j(n)^* x_i(n+e_j)^* x_j(n+e_i) x_i(n),\; i2$, we have to refer to Proposition~(\ref{moduli}). \\
We consider now the expectation
value $\tr(L^4)$ of the Wilson Hamiltonian $H(x)=(L_x^4)_{00}$ and show that it
is minimized by a gauge field which has constant curvature
$F_{ij}=-1$. This minimization holds in the abelian case among all
fields $F$ with $dF=0$ and in dimension $2$ among all fields $F$.
Unless in the flux-phase problem which deals with $\tr|L|$ (and which is
settled in finite dimensional, two dimensional situations),
the calculations for $\tr(L^4)$ is easy:
\begin{propo}
The expectation value $\tr(L^4)$ of the Wilson Hamiltonian
takes values in the interval
$[4d^2+2d,12 d^2- 6d)]$. The minimum is achieved for $F=-1$ and the maximum
for $F=1$.
\end{propo}
\begin{proof}
$\tr(L^4)$ takes values in the interval
$[A - 2 B,A]$,
where $A=4d^2+ 2d (2d-1) + 4 d(d-1)$ is the number of closed paths of length $4$
starting at $0 \in \ZZ^d$ and $B=4 d(d-1)$ is the number of plaquettes
at $0$, where plaquettes with different orientation are considered as
different.
$F=-1$ gives $A-2B$ and $F=1$ gives
$A$ and no smaller or bigger values can occur.
\end{proof}
The problem to find the extrema of $\tr(L^6)$ is more difficult
because loops over two plaquettes and so correlations occur.
In two dimensions, where it is possible to
minimize over all fields $F$ instead to minimize over the
gauge potential,
we have to find a minimum on $M(X_2)=M(U^{\ZZ^2})$ of
the functional
$$ \mu \mapsto A + B \int {\rm Trace}(x(0)) \; d\mu(x)
+ C/2 \int {\rm Trace}(x(0) x(e_1) + x(0) x(e_2)) \; d\mu(x) \; , $$
where $A$ is the number of paths of length $6$ starting at $n=0$,
giving zero winding number to all plaquettes, $B=128$ is the number of
such paths winding around one plaquette and $C=16$ is the number of such
paths winding around two
plaquettes. In the abelian case $U=U(1)$ and if we
consider the much simpler one dimensional
problem of a constant field $F=e^{i \alpha}$,
we have to minimize
$$ A + B \cos(\alpha) + C \cos(2 \alpha) \; . $$
Since $B>4C$,
the minimum is $\alpha=\pi$ which corresponds to a constant
field $F=-1$.
\section{A random walk expansion}
The functionals $I,I_n,\overline{I}$ can not be expected to be continuous
in $\mu$.
(The question seems already to be unsettled in the case
$d=2$, $N=1$ and constant curvature
$e^{2\pi i \alpha}$, with irrational $\alpha$, where
$dk$ is the density of states of the almost Mathieu operator and where
the unknown values $I_1,I_2$ are believed to satisfy $I_1=I_2=0$.)
It is therefore convenient to consider
for every $\beta \in \CC$ the functional
$$ \mu \mapsto
I_{n,\beta}=- \int \log|E+\beta| \; dk^{(n)}_{\mu}(E)
= - \log|\det_{\mu}(L^{(n)}+\beta)|\; $$
which has the property that $I_{n,\beta}+\log(\beta)$
is real analytic in $\beta$ and continuous in $\mu$ for $|\beta|>2dn$.
It is convenient to consider also $I_{n,z}(\mu)= -\log |\det_{\mu}(1-zL)|$
which is real analytic in
a neighborhood of $0$. Note that by the same arguments given below,
the functionals $I_{n,\beta}$ rsp. $I_{n,z}$ have all minimizers.
By an analytic continuation, the knowledge of the Taylor coefficients
of the superharmonic function
$\beta \mapsto I_{n,\beta}+\log(\beta)$ in a neighborhood of $\infty$ determines
$I_n=I_{n,\beta=0}$.
The Taylor coefficients can be computed in terms of the gauge fields.
For sake of simplicity, we consider first the functional $I$.
\begin{propo}
For $|\beta|>4d$, the Taylor expansion
$$ I_{\beta}(\mu)+\log(\beta)
= \sum_{m=1}^{\infty}
\left(
\sum_{l=0}^{m}
\left( \begin{array}{c} m \\ l \\ \end{array} \right)
\tr_{\mu} (L^l) \tr_{\mu} (L^{m-l})
\right) \frac{\beta^{-m}}{m} \; $$
holds. The functions $\mu \mapsto \tr_{\mu}(L^n)=\int E^n \; dk_{\mu}(E)$ are
nonvanishing only for even $n$ and satisfy
$$ \tr_{\mu}(L^n)
= \int_X \sum_{\gamma \in \Gamma_n} {\rm Trace} (\int_{\gamma} x)\; d\mu(x) \;, $$
where $\Gamma_n$ is the set of all closed paths in $\ZZ^d$ of length $n$,
${\rm Trace}$ is the normalized trace in $U(N)$
and $\int_{\gamma} x$ is the product
$x_{\gamma(n)-\gamma(n-1)}(\gamma(n-1)) x_{\gamma(n-1)-\gamma(n-2)}(\gamma(n-2))
\dots x_{\gamma(2)-\gamma(1)}(\gamma(1))$ of the gauge field along the path
$\gamma$ (and where $x_{e_i}=x_i$).
\end{propo}
\begin{proof}
$$ I_{\beta}(\mu) = - \int \int \log|1-\frac{(E'+E)}{\beta}|
\; dk_{\mu}(E) \; dk_{\mu}(E') - \log(\beta) \; . $$
An expansion of the logarithm gives
\begin{eqnarray*}
I_{\beta}(\mu) + \log(\beta)
&=& \sum_{m=1}^{\infty} \int \int (E+E')^m \; dk_{\mu}(E) \; dk_{\mu}(E')
\frac{\beta^{-m}}{m} \\
&=& \sum_{m=1}^{\infty} \int \tr_{\mu}((L+E')^m) \; dk_{\mu}(E')
\frac{\beta^{-m}}{m} \\
&=& \sum_{m=1}^{\infty} \sum_{k=0}^m
\left( \begin{array}{c} m \\ l \\ \end{array} \right)
\int \tr_{\mu}(L^k) (E')^{m-k} \; dk_{\mu}(E') \frac{\beta^{-m}}{m}
\end{eqnarray*}
which gives the expression in the proposition.
The random walk computation of $\tr(L^n)$ follows from
the computation of the trace of each summand in
$L^n=(x_1+x_1^* + x_2+x_2^* \dots + x_d+x_d^*)^n$.
\end{proof}
We see from the development that
$\mu \mapsto I_{\beta}(\mu)$ behaves "quadratically" in $\mu$.
For the determinant of
$L^{(n)}$, a similar result holds and
$\mu \mapsto I_{\beta,n}(\mu)$ behaves as a "polynomial" of
degree $n$:
\begin{coro}
For $|\beta|>2nd$, the Taylor expansion
$$ - \log \det(L^{(n)}+\beta) + \log(\beta)
= \sum_{m=1}^{\infty}
\left(
\sum_{l_1+ \dots + l_n = m}
\tr_{\mu} (L^{(l_1)}) \cdots \tr_{\mu} (L^{(l_n)})
\right) \frac{\beta^{-m}}{m} \; $$
holds, where $\tr_{\mu}(L^{(0)})=1$.
\end{coro}
\begin{proof}
The proof is easily adapted from the above proof, using
$$ \int_{\RR^n} (E_1 + E_2 \dots + E_n)^m \; dk_{\mu}(E_1) \dots dk_{\mu}(E_n)
= \sum_{l_1+ \dots + l_n = m}
\tr_{\mu} (L^{(l_1)}) \cdots \tr_{\mu} (L^{(l_n)}) \; . $$
\end{proof}
Remarks. \\
1) Since $\tr(L^2)= 2d$, the first interesting term is
$\tr(L^4)$. For large $|\beta|$, an approximation
of $I_{\beta}(\mu)=I_{2,\beta}(\mu)$
is given by
$$ \log(\beta) + 2 (2d) \beta^{-2} +
(6 (2d)^2+4 \tr(L^4)) \beta^{-4}
+ O(\beta^{-6}) \; . $$
2) The knowledge of the function $\beta \mapsto I_{\beta}$ for fixed $\mu$
does clearly not determine $\mu$ in general, because of gauge freedom.
However, the function $\beta \mapsto I_{2,\beta}$ determines the moments
$\tr(L^n)$ of the density of states of $L=L_{\mu}$
and so the density of states $dk_{\mu}$ itself.
The reason is that a sequence $a_k,k=0, \dots $ of real
numbers is determined from the sequence $b_m=\sum_{k=0}^m
\left( \begin{array}{c} m \\ k \\ \end{array} \right) a_k a_{m-k}, m=0,1 \dots$
and the sign of $a_0$. \\
3) The function $\log \det(L-\beta)$ has not an an analytic continuation
to the resolvent set of $L$ and this is why the real part is usually
considered.
However, $\beta \mapsto \log \det(L-\beta)$
is a Herglotz function on ${\rm Im}(E)>0$ and the imaginary
part on the real axes is proportional to the integrated density of states. The
determinant $\beta \mapsto \det(L-\beta)$
is defined everywhere on $\CC$ and analytic
on the resolvent set of $L$. \\
In dimension $d=2$, and $U(1)$, there are some cases, where more can be said
(see \cite{Kni95}):\\
1) If $L=L_{{\rm free}}$, then
$$ \det(1-zL)= \sum_{k=1}^{\infty} \frac{z^k}{k} |\Gamma_k| \; , $$
where $\Gamma_k$ is the set of closed paths in $\ZZ^d$ of length $k$
starting at $0 \in \ZZ^d$ and where
$|\Gamma_k|$ denotes its cardinality. \\
2) If $L$ has curvature constant to $-1$, then
$$ \det(1-zL)= \sum_{k=1}^{\infty} \frac{z^k}{k}
\sum_{\gamma \in \Gamma_k} (-1)^{n(\gamma)} \; , $$
where $n(\gamma)$ is the sum of the winding numbers $n(\gamma,P)$
over all plaquettes $P$ in
$\ZZ^2$, which is well defined because $n(\gamma,P) \neq 0$ only for
finitely many $n$. \\
3) A generalisation is when the curvature is constant
$e^{2 \pi i \alpha}$. Then
$$ \det(1-zL)= \sum_{k=1}^{\infty} \frac{z^k}{k}
\sum_{\gamma \in \Gamma_k} \prod_P e^{2\pi i n(\gamma,P)} \; , $$
where $n(\gamma,P)$ is the winding number of the path $\gamma$ with respect
to a point in the plaquette $P$. \\
If $\alpha$ is irrational, then the density of states of $L$ is the same
as the density of states of the one dimensional almost Mathieu operator
$(K_{\theta} u)_n = u_{n+1} + u_{n-1} + 2 \cos(\theta+2\pi \alpha n) u_n$
on $l^2(\ZZ,\CC)$ (see \cite{Shu94,Las95,Jit95} for reviews).
This is not true for rational $\alpha$, where
the spectrum of $K_{\theta}$ depends on $\theta \in \TT^1$. \\
4) If $\mu$ is the Haar measure on $X_1=(U(1) \times U(1))^{\ZZ^2}$, then
$$ \det(1-zL)= \sum_{k=1}^{\infty} \frac{z^k}{k}
|\Gamma_k^0| \; , $$
where $\Gamma_k^0$ is the subset of those $\gamma \in \Gamma_k$,
which give zero winding numbers to all plaquettes. \\
5) If $\mu$ is the Haar measure on $(\ZZ_p \times \ZZ_p)^{\ZZ^2}$, then
$$ \det(1-zL)= \sum_{k=1}^{\infty} \frac{z^k}{k}
|\Gamma_k^p| \; , $$
where $\Gamma_k^p$ is the subset of $\gamma \in \Gamma_k$,
which give modulo $p$ a zero winding number to all plaquettes.
\section{Relation with a dynamical zeta function}
Consider the following general definition of a dynamical zeta function
(see \cite{Ruelle2} p.1). Given a dynamical system $(M,f)$, where
$M$ is a homeomorphism of the topological space $M$. Assume the set of
fixed points ${\rm Fix}(f^m)$ is finite for all $m \in \NN$.
If $A: M \rightarrow M(N,\CC)$ is a continuous matrix-valued function,
define $\zeta(z)$ by
$$ \log \zeta(z) = \sum_{k=1}^{\infty} \frac{z^k}{k} \sum_{m \in {\rm Fix}(f^k)}
{\rm Trace}(\prod_{j=0}^{k-1} A(f^k m) \; . $$
Consider now the set $B$ consisting of the $2d$ bonds connecting $0$ with
its neighbors in $\ZZ^d$. Define
$M=B^{\ZZ} \times \ZZ^d$ and the map $f=(d,b) = (\sigma(d),b + d_0) $,
where $\sigma$ is the shift on $B^{\ZZ}$. Every $x \in X$
defines a map $A_x$ on $M$ by $A_x(d,b)= x_b(d)$, where $x_{-l}$ is
interpreted as $x_l^*$. The map $f^k$ has finitely many fixed points
on the set $\{(d,b) \; | \; b=0 \}$ and these $k$- periodic points of $f$
correspond bijectively to closed paths in $\ZZ^d$ of length $k$ starting
at $n=0$. Averaging
$$ \sum_{k=1}^{\infty} \frac{z^k}{k}
\sum_{m \in {\rm Fix}(f^k) \cap \{b=0\}}
{\rm Trace}( \prod_{j=0}^{k-1} A_x(f^k m) ) \; $$
over $X$ with respect to the measure $\mu$
gives $\log \det(1-zL)$.
\section{A relation with the entropy functional}
Given a compact metric space $X$, $d$ commuting homeomorphisms $T_i$
on $X$ and $d$ continuous maps $A_i: X \rightarrow U$.
For every shift invariant measure, we have the random operator
$L_x=A_x+A_x^*+V_x$ with $A_x u(n) = \sum_i U_i(T^nx) u(n+e_i)$
and $V_x u(n) =V(T^nx) u(n)$.
Again, the potential theoretical energy of the density of states is
a lower semicontinuous functional on the compact set of $T_i$-invariant
measures and the minimum is attained. \\
Consider now the case $d=1$ and a smooth symplectic
twist map $T : \TT^{2n} \rightarrow \TT^{2n}$ defined by
$$ T: (q,p) \mapsto (2q-p-f(q),q) \; $$
with smooth $f: \TT^n \rightarrow \RR^n$.
In the case $n=1$, $f(x)=\gamma \sin(x)$, this is the Standard map.
For each $T$-invariant measure $\mu$, we have
an ergodic Schr\"odinger operator $L=L(\mu)$ for which we can form
the finite determinant $\det L$. The calculation of this determinant
is in general difficult and it is in general not known,
whether $|\det_{\mu} L|>1$ or $|\det_{\mu} L|=1$ for the Lebesgue measure
$\mu$ on $\TT^{2n}$ because of the following consequence of the
Thouless formula.
\begin{lemma}
$n \cdot \log |\det_{\mu} L|$ is the metric entropy of the invariant
measure $\mu$ of twist map $T$ if $\mu$ is absolutely continuous
with respect to Lebesgue measure.
\end{lemma}
\begin{proof}
The Thouless formula is
$n \cdot {\rm tr}(\log|L|) = \sum_{i=1}^n \lambda_i(0)$, where
$\lambda_i(0)$ are the $n$ largest Lyapunov exponents
of the transfer cocycle $A_E$ of $L$ for $E=0$.
But $A$ is in the same time also the Jacobean $DT$ of the twist map $T$.
Pesin's formula (see \cite{Mane})
states that the Lyapunov exponent is the metric entropy
of the invariant measure $\mu$, if $\mu$ is absolutely continuous with
respect to the Lebesgue measure.
\end{proof}
Maximizing the functional $|\det L|$ for Schr\"odinger operators
arrizing in a monotone twist (see \cite{KnLa95} for spectral results)
is related to the metric entropy problem.
\begin{coro}
If the maximum $\mu$ of $\mu \mapsto |\det_{\mu}(L)|$
is absolutely continuous with respect
to Lebesgue measure on $\TT^2$, then $\log|\det_{\mu}(L)|$ is the topological
entropy of the twist map.
\end{coro}
\begin{proof}
If $\mu$ is absolutely continuous, then $\log|\det_{\mu} L|$ is a metric entropy
$h_{\mu}(T)$ and so by the variational principle
$h_{top}(T) \geq \log|\det_{\mu} L|$.
Ruelle's inequality shows that
$h_{\mu}(T) \leq \log | \det_{\mu} L|$ for all invariant measures.
The maximal value of $\mu \mapsto \log|\det_{\mu} L|$
is therefore also bigger or equal than the topological entropy $h_{top}(T)$.
\end{proof}
\section{Questions}
We would like to know explicit
minima of the potential theoretical energy of the density
of states. Is there a unique gauge invariant minimum? If yes,
it would be ergodic.
A natural guess for a unique gauge invariant
ergodic minimum is the Haar measure $\mu$ on $X$. If $U$ is abelian and $d=2$,
we can in this case compute the moments of the density of states but we
could not determine $|\det_{\mu} L|$.
We also do not know if we can get the maximal value $0$
for the energy in higher dimensions nor do we know the minima of the functionals
$\tr_{\mu}(L^{2n})$ for $n>2$. \\
Is always $I_{n+1}(L) \leq I_n(L)$? We know only that it is true for large $n$,
since we can decide, how the sequence $I_n(L)$ converges to $-\infty$.
The monotonicity would have consequences like
that the invertibility of $L$,
which implies $I_1(L)<0$, would give $I_2(L)<0$ and so imply that
the capacity of the spectrum is bigger than one. \\
In all the examples of operators attached to gauge fields we know,
$L$ is not invertible: examples are
the Hofstadter case $d=2$ with constant rational field $e^{2\pi i p/q}$, the
Haar measure case, the free case. Does there exist $\mu \in M(X)$ such that
$L_{\mu}$ is invertible? \\
In the finite dimensional case, there
exists a natural Haar measure $\rho$ on
all operators $L$ and so a natural partition function
$\int e^{-\beta I(L)} \; d\rho(L)= \int |\det_{\mu} L^{(2)}| \; d\rho(\mu)$.
The largest contribution to this
partition function is expected to come
from the minimizers $\mu$, especially if the inverse temperature
$\beta$ is large. The determination of a natural measure $\rho$ on $Y$
or $X_2$ and the computation of the partition function is
the quantum version of the variational problem to minimize the energy.
It can be shown with
the Ryll-Nardzewski fixed point theorem
that on any compact metric space $Y$,
there exist measures $\rho$ which are invariant under all isometries.
Such a measure $\rho$ can be chosen in such a way that it gives positive measure
to every open set.
%because else, the support of the measure is invariant
%under all isometries and so also the complement, the closure of the
%complement which is itself a compact set cariing a measure invariant
%under all isometries.
Moreover, with such a measure $\rho$ on $M(X_2)$, the knowledge of the
function $\beta \mapsto
\int e^{-\beta I(\mu)} \; d\rho(\mu)$ determines the minimal value of $I$.
Uniqueness of $\rho$ can only be expected if the group of isometries
is transitive on $Y$.
Whatever measure $\rho$ is taken, we expect from the asymptotic behavior of
$I_n$ that the partition function satisfies asymptotically
$\int e^{-I_n(\mu)} \; d\rho(\mu) \sim e^{-\gamma/2}/\sqrt{n/2}$.
\section*{Appendix A: The Thouless formula for Schr\"odinger operators
on the strip with off diagonal elements}
Given a measure preserving invertible
transformation $T$ on a probability space $(X,\mu)$.
Given $a,b \in L^{\infty}(X,M(N,\RR))$ so that
$a^{-1} \in L^{\infty}(X,M(N,\RR))$. Consider the random operator
$x \mapsto L_x$ on $\Hcal=l^2(\ZZ,\CC^N)$ given by
$$ (L_x u)_n = a_n u_{n+1} + a_{n-1}^* u_{n-1} + b_n u_n \; , $$
where $a_n = a(T^nx), b_n=b(T^nx)$.
Denote by $dk$ the density of states of $L$ defined by the functional
$f \mapsto \tr(f(L)) = \int {\rm Trace}(f(L_x))_{00} \; d\mu(x)$, where
${\rm Trace}$ is the normalized trace on $M(N,\RR)$ satisfying ${\rm Trace}(1)=1$.
Define $\lambda(E)= \int \log|E-E'| \; dk(E')$. Denote by
$\lambda_i(E), i=1, \dots, N$
the nonnegative Lyapunov exponents of the $2N \times 2N$ transfer cocycle
$$ A_E(x) = \left( \begin{array}{cc} E-b(x) & -a(T^{-1}x)^{2} \\
1 & 0 \\
\end{array} \right) a(T^{-1}x) \; , $$
which has the property that
$$ A_{n,E}= \left( \begin{array}{cc} E-b_n & -a_{n-1}^{2} \\
1 & 0 \\
\end{array} \right) a_{n-1}^{-1} \; $$
satisfies $A_{n,E} v_n = v_{n+1}$ if $L u=E u$ and
$v_n = (a_n u_{n+1}, u_n)$. Denote by $\lambda_i(a)$ the $N$ Lyapunov exponents
of the cocycle $x \mapsto a(x)$.
\begin{propo}
The Thouless formula for random Jacobi matrices on the strip is
$$ \lambda(E) = N^{-1} \sum_{i=1}^N (\lambda_i(A_E) + \lambda_i(a)) \; , $$
where $\lambda_i(A_E)$ are the $N$ largest Lyapunov exponents of the
$SL(2N,\CC)$ transfer cocycle $A_E$ and
$\lambda_i(a)$ are the $N$ Lyapunov exponents of
the $GL(N,\CC)$ cocycle $a$.
\end{propo}
\begin{proof}
The proof is a straightforward generalisation from the proofs
\cite{Cycon,Carmona,KoSi88} in the
known situations (see examples 1),2) below): \\
It suffices to prove the formula for ${\rm Im}(E) \neq 0$, where $\lambda(E)$
is analytic. The usual subharmonicity argument assures then that the formula
is true everywhere on $\CC$. By ergodic decomposition it suffices also
to assume that $T$ is ergodic. Define $a^n(x)=a_{n-1}(x) \cdots a_2(x) a_1(x)$.
We have
$$ A^n_E(x)=A_{n,E}(x) \cdots A_{1,E}(x)=\left( \begin{array}{cc}
P^n_E(x) & Q^{n-1}_E(x) \\
a(T^{n-1}) P^{n-1}_E(x) & a(T^{n-1}) Q^{n-2}_E(x) \\
\end{array} \right) \; , $$
where $E \mapsto P_E^n(x),Q_E^n(x)$
are matrix coefficient polynomials of degree $n$ of the form
$$ P_E^n(x) = a^n(x) \prod_{k=1}^n (E-E_j^{(n)}(x)), \;
Q_E^n(x) = a^n(x) \prod_{k=1}^n (E-F_j^{(n)}(x)) \; $$
Then, both $(Nn)^{-1}\log|\Det P^n_E(x)|$
and $(Nn)^{-1}\log|\Det Q^n_E(x)|$, (here $\Det$ is the
finite dimensional determinant),
converge almost everywhere
to the average of the largest $N$ Lyapunov exponents of $A_E$. Also
$(N n)^{-1} \log |\det(a^n)(x)|$ converges to the average of the $N$
Lyapunov exponents of $a$. (Oseledec's theorem
assures that $(a^n(x) (a^n)^*(x))^{1/2n}$ converges for almost all $x$ to a constant
matrix with the $N$ eigenvalues $\exp(\lambda_i(a))$.)
Because $E_j^{(n)}(x)$ (rsp. $F_j^{(n)}(x)$) are
eigenvalues of $L_x$ with Dirichlet boundary conditions $u_0=u_n=0$, (rsp.
$u_1,u_{n+1}=0$), we know by the Avron-Simon lemma, that both
$(N n)^{-1} \log \prod_{k=1}^n (E-E_j^{(n)})$ and
$(N n)^{-1} \log \prod_{k=1}^n (E-E_j^{(n)})$ converges to
$\int \log|E-E'| \; dk(E')$.
\end{proof}
Examples. \\
1) If $N=1$, Birkhoff's ergodic theorem gives
$\lambda_1(a) = \int \log|a| \; d\mu$ and gives the known special case
$\lambda(E) = \lambda(A_E) + \int \log|a| \; d\mu$. \\
2) If $a=1$, then $\lambda_i(a)=0$ and we have the known special case
$\lambda(E) = N^{-1} \sum_{i=1}^N \lambda_i(A_E)$.
(The formulas in
\cite{KoSi88,Carmona} differ by a factor $N$ which is due to a different
${\rm Trace}$ on finite dimensional matrices. \\
3) If $a,b$ take values in diagonal matrices and $a_i$ are the diagonal
functions, then $L$ is a direct sum of $N$ ordinary Jacobi operators $L_i$
with density of states $dk_i$ and $dk$ is the average of the $dk_i$
and $\lambda(E)$ is therefore the average of the functions
$\lambda_i(E)=\int \log|E-E'| \; dk_i(E')$.
The $N$ nonnegative Lyapunov exponents of $A_E$ are the Lyapunov exponents
of the transfer matrices $A_{E,i}$ of the operators $L_i$ and the
$N$ Lyapunov exponents of $a$ are given by
$\lambda_i(a)=\int \log|a_i| \; d \mu$ so that
$N^{-1} \lambda_i(E) = N^{-1} \sum_{i=1}^N (\lambda_i(A_E) + \lambda_i(a))$
follows from the one dimensional case.
\section*{Appendix B: A result of Parthasarthy}
Parthasarthy \cite{Par61} proved that if $T$ is the shift on the product
space $(X=M^{\ZZ},\Bcal^{\ZZ})$,
where $M$ is a compact metric space equiped with the Borel $\sigma$-algebra
$\Bcal$, then the set $P$ of periodic shift invariant
measures is dense in the set $E$ of all ergodic measures
which is itself dense
in the set $M$ of all shift invariant probability
measures. Parthasarthy's original proof
in the special case $d=1$ generalizes
directly to higher dimensions:
\begin{lemma}[Parthasarthy]
$P$ is dense in $E$ which is dense in $M$.
\end{lemma}
\begin{proof}
(i) We first show that $E$ is dense in $M$.
Partition $\ZZ^d$ into cubes $\Lambda_r^n=\prod_{i=1}^d [r_i-n,r_i+n]$ with
$r \in \ZZ^d$. Let $\Acal_r^n$ by the sub $\sigma$-algebra of $\Acal$
generated by functions depending only on coordinates in $\Lambda_r^n$.
Given $\mu \in M$, let $\mu_r^n$ be the restriction of $\mu$
to $\Acal_r^n$. Define $\nu_n=\prod_{r \in \ZZ^d} \mu_r^n$ which is defined
on $\Acal$ and invariant and ergodic under the shifts $T_i^{2n+1}$.
Define $\mu_n=(|\Lambda_0^n|)^{-1} \sum_{k \in \Lambda_0^n} \nu_n(T^k)$
which is invariant and ergodic for $T$.
Since $\mu_n$ agrees with $\nu_n$ on $\Acal_0^n$ and if $A \in \Acal_0^k$
then $\nu_n(T^rA)=\mu(A)$ for $A \in \Acal_0^n$, we get for fixed $k$
$$ |\mu_n(A)-\mu(A)| \leq \frac{1}{|\Lambda_0^n|}
\sum_{j \in \Lambda_0^n} |\nu_n(T^j A) - \mu(A) |
\leq \frac{2d |\Lambda_0^k|}{|\Lambda_0^n|} \rightarrow 0 $$
so that $\mu_n(A) \rightarrow \mu(A)$ for $A \in \Acal_0^k$. Since this is true
for every $k$, one has $\mu_n(A) \rightarrow \mu(A)$ for all $A \in \Acal$. \\
(ii) Given $\mu \in E$, there exists $x \in X$ such that
$\mu_n = (1/|\Lambda_0^n|) \sum_{k \in \Lambda_0^n} \delta(T^k x)$
converges to $\mu$. Let $y$ be a point in $X$ which is $2n+1$ periodic
for each $T_i$ and which coincides with $x$ on $\Lambda_0^n$. Define
$\nu_n = (1/|\Lambda_0^n|) \sum_{k \in \Lambda_0^n} \delta(T^k y)$.
which is in $P$. The same argument as in (i) shows that
$\mu_n(A) \rightarrow \nu_n(A)$ for $A \in \Acal_0^k$ and so
$ \nu_n \rightarrow \mu$.
\end{proof}
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