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M. Shcherbina

Publications and source records attributed to M. Shcherbina.

15 recordsLinked to original sources
arXiv2018

Szegö-type Theorems for One-Dimensional Schrodinger Operator with Random Potential (smooth case)

The paper is a continuation of work [15] in which the general setting for analogs of the Szegö theorem for ergodic operators was given and several interesting cases were considered. Here we extend the results of [15] to a wider class of test functions and symbols which determine the Szegö-type asymptotic formula for the one-dimensional Schrodinger operator with random potential. We show that in this case the subleading term of the formula is given by a Central Limit Theorem in the spectral context, hence the term is asymptotically proportional to $L^{1/2}$, where $L$ is the length of the interval on which the Schrodinger operator is initially defined. This has to be compared with the classical Szegö formula, where the subleading term is bounded in $L$, $L \to \infty$. We prove an analog of standard Central Limit Theorem (the convergence of the probability of the corresponding event to the Gaussian Law) as well as an analog of the almost sure Central Limit Theorem (the convergence with probability 1 of the logarithmic means of the indicator of the corresponding event to the Gaussian Law). We illustrate our general results by establishing the asymptotic formula for the entanglement entropy of free disordered Fermions for non-zero temperature.

math-ph↗
arXiv2016

Large Block Properties of the Entanglement Entropy of Disordered Fermions

We consider a macroscopic disordered system of free $d$-dimensional lattice fermions whose one-body Hamiltonian is a Schrödinger operator $H$ with ergodic potential. We assume that the Fermi energy lies in the exponentially localized part of the spectrum of $H$. We prove that if $S_Λ$ is the entanglement entropy of a lattice cube $Λ$ of side length $L$ of the system, then for any $d \ge 1$ the expectation $\mathbf{ E}\{L^{-(d-1)}S_Λ\}$ has a finite limit as $L \to \infty$ and we identify the limit. Next, we prove that for $d=1$ the entanglement entropy admits a well defined asymptotic form for all typical realizations (with probability 1) as $ L \to \infty$. According to numerical results of [33] the limit is not selfaveraging even for an i.i.d. potential. On the other hand, we show that for $d \ge 2$ and an i.i.d. random potential the variance of $L^{-(d-1)}S_Λ$ decays polynomially as $L \to \infty$, i.e., the entanglement entropy is selfaveraging.

quant-ph↗
arXiv2010

Orthogonal and symplectic matrix models: universality and other properties

We study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measure. For these models we find the bounds (similar to the case of hermitian matrix models) for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order O(1). We prove also universality of local eigenvalue statistics in the bulk.

math-ph↗
arXiv2010

Fluctuations of eigenvalues of matrix models and their applications

We study the expectation of linear eigenvalue statistics of matrix models with any $β>0$, assuming that the potential $V$ is a real analytic function and that the corresponding equilibrium measure has a one-interval support. We obtain the first order (with respect to $n^{-1}$) correction terms for the expectation and apply this result to prove bulk universality for real symmetric and symplectic matrix models with the same $V$.

math-ph↗
arXiv2009

Central limit theorem for fluctuations of linear eigenvalue statistics of large random graphs

We consider the adjacency matrix $A$ of a large random graph and study fluctuations of the function $f_n(z,u)=\frac{1}{n}\sum_{k=1}^n\exp\{-uG_{kk}(z)\}$ with $G(z)=(z-iA)^{-1}$. We prove that the moments of fluctuations normalized by $n^{-1/2}$ in the limit $n\to\infty$ satisfy the Wick relations for the Gaussian random variables. This allows us to prove central limit theorem for $\hbox{Tr}G(z)$ and then extend the result on the linear eigenvalue statistics $\hbox{Tr}ϕ(A)$ of any function $ϕ:\mathbb{R}\to\mathbb{R}$ which increases, together with its first two derivatives, at infinity not faster than an exponential.

math-ph↗
arXiv2007

On Universality for Orthogonal Ensembles of Random Matrices

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.

math-ph↗
arXiv2007

Bulk Universality and Related Properties of Hermitian Matrix Models

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally $C^{2}$ and locally $C^{3}$ function (see Theorem \ref{t:U.t1}). The proof as our previous proof in \cite{Pa-Sh:97} is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the $sin$-kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper \cite{BPS:95} on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.

math-ph↗
arXiv2007

Double scaling limit for matrix models with non analytic potentials

We study the double scaling limit for unitary invariant ensembles of random matrices with non analytic potentials and find the asymptotic expansion for the entries of the corresponding Jacobi matrix. Our approach is based on the perturbation expansion for the string equations. The first order perturbation terms of the Jacobi matrix coefficients are expressed through the Hastings-McLeod solution of the Painleve II equation. The limiting reproducing kernel is expressed in terms of solutions of the Dirac system of differential equations with a potential defined by the first order terms of the expansion.

cond-mat.dis-nn↗
arXiv2006

Dynamical behavior of a large complex system

Limit theorems for a linear dynamical system with random interactions are established. These theorems enable us to characterize the dynamics of a large complex system in details and assess whether a large complex system is stable or unstable, answering an open question raised 30 years ago in the literature

math-ph↗
arXiv2006

Stability of the dynamics of an asymmetric neural network

We study the stability of the dynamics of a network of n neurons intercting linearly through a random gaussian matrix of excitatory and inhibitory type. Using the aproach developed in a previous paper we show some interesting properties of the dynamic of this system for large values of n. We got sufficient conditions for getting diverging synchronized behavior or stability.

math-ph↗
arXiv2003

On the edge universality of the local eigenvalue statistics of matrix models

Basing on our recent results on the $1/n$-expansion in unitary invariant random matrix ensembles, known as matrix models, we prove that the local eigenvalue statistic, arising in a certain neighborhood of the edges of the support of the Density of States, is independent of the form of the potential, determining the matrix model. Our proof is applicable to the case of real analytic potentials and of supports, consisting of one or two disjoint intervals.

math-ph↗
arXiv2001

Rigorous Solution of the Gardner Problem

We prove rigorously the well-known result of Gardner about the typical fractional volume of interactions between N spins which solve the problem of storing a given set of random patterns. The Gardner formula for this volume in the limit N,p \to \infty, p/N \to αis proven for all values of α. Besides, we prove a useful criterion of the factorisation of all the correlation functions for a class of spin glass model.

math-ph↗
arXiv2000

On the Critical Capacity of the Hopfield Model

We estimate the critical capacity of the zero-temperature Hopfield model by using a novel and rigorous method. The probability of having a stable fixed point is one when $α\le 0.113$ for a large number of neurons. This result is an advance on all rigorous results in the literature and the relationship between the capacity $α$ and retrieval errors obtained here for small $α$ coincides with replica calculation results.

math-ph↗