arXiv subjects

L. Pastur

Publications and source records attributed to L. Pastur.

At least 19 recordsLinked to original sources

arXiv2024

Entanglement Entropy of Free Fermions with a Random Matrix as a One-Body Hamiltonian

We consider a quantum system of large size $N$ and its subsystem of size $L$ assuming that $N$ is much larger than $L$, which can also be sufficiently large, i.e., $1 \ll L \lesssim N $. A widely accepted mathematical version of this heuristic inequality is the asymptotic regime of successive limits: first the macroscopic limit $N \to \infty$, then an asymptotic analysis of the entanglement entropy as $L \to \infty$. In this paper, we consider another version of the above heuristic inequality: the regime of asymptotically proportional $L$ and $N$, i.e., the simultaneous limits $L \to \infty,\; N \to \infty, L/N \to \lambda >0$. Specifically, we consider the system of free fermions which is in its ground state and such that its one-body Hamiltonian is a large random matrix, that is often used to model the long-range hopping. By using random matrix theory, we show that in this case, the entanglement entropy obeys the volume law known for systems with short-ranged hopping but described either by a mixed state or a pure strongly excited state of the Hamiltonian. We also give a streamlined proof of Page's formula for the entanglement entropy of the black hole radiation for a wide class of typical ground states, thereby proving the universality of the formula.

quant-ph↗

arXiv2020

On Random Matrices Arising in Deep Neural Networks: General I.I.D. Case

We study the distribution of singular values of product of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the population covariance matrices assumed to be non-random or random but independent of the random data matrix in statistics and random matrix theory are now certain functions of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [25, 13] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability has to be justified. The justification has been given in [22] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper we use another, more streamlined, version of the techniques of random matrix theory to generalize the results of [22] to the case where the entries of the synaptic weight matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, extends the property of the so-called macroscopic universality on the considered random matrices.

math-ph↗

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↗

arXiv2017

The absence of the selfaveraging property of the entanglement entropy of disordered free fermions

We consider the macroscopic system of free lattice fermions in one dimension assuming that the one-body Hamiltonian of the system is the one dimensional discrete Schrödinger operator with independent identically distributed random potential. We show analytically and numerically that the variance of the entanglement entropy of the segment $[-M,M]$ of the system is bounded away from zero as $M\rightarrow \infty $. This manifests the absence of the selfaveraging property of the entanglement entropy in our model, meaning that in the one-dimensional case the complete description of the entanglement entropy is provided by its whole probability distribution. This also may be contrasted the case of dimension two or more, where the variance of the entanglement entropy per unit surface area vanishes as $% M\rightarrow \infty $ \cite{El-Co:17}, thereby guaranteeing the representativity of its mean for large $M$ in the multidimensional case.

quant-ph↗

arXiv2017

The absence of the selfaveraging property of the entanglement entropy of disordered free fermions in one dimension

We consider the macroscopic system of free lattice fermions in one dimensions assuming that the one-body Hamiltonian of the system is the one dimensional discrete Schrödinger operator with independent identically distributed random potential. We show that the variance of the entanglement entropy of the segment $[-M,M]$ of the system is bounded away from zero as $% M\rightarrow \infty $. This manifests the absence of the selfaveraging property of the entanglement entropy in our model, meaning that in the one-dimensional case the complete description of the entanglement entropy is provided by its whole probability distribution. This also may be contrasted the case of dimension two or more, where the variance of the entanglement entropy per unit surface area vanishes as $M\rightarrow \infty $ \cite{El-Co:17}, thereby guaranteing the representativity of its mean for large $M$ in the multidimensional case.

quant-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↗

arXiv2014

On the Area Law for Disordered Free Fermions

We study theoretically and numerically the entanglement entropy of the $d$-dimensional free fermions whose one body Hamiltonian is the Anderson model. Using basic facts of the exponential Anderson localization, we show first that the disorder averaged entanglement entropy $\langle S_Λ\rangle$ of the $d$ dimension cube $Λ$ of side length $l$ admits the area law scaling $\langle S_Λ\rangle \sim l^{(d-1)}, \ l \gg 1$ even in the gapless case, thereby manifesting the area law in the mean for our model. For $d=1$ and $l\gg 1$ we obtain then asymptotic bounds for the entanglement entropy of typical realizations of disorder and use them to show that the entanglement entropy is not selfaveraging, i.e., has non vanishing random fluctuations even if $l \gg 1$.

quant-ph↗

arXiv2013

The Central Limit Theorem for Linear Eigenvalue Statistics of the Sum of Independent Matrices of Rank One

We consider $n\times n$ random matrices $M_{n}=\sum_{α=1}^{m}{τ_{α}}\mathbf{y}_{α}\otimes \mathbf{y}_{α}$, where $τ_{α}\in \mathbb{R}$, $\{\mathbf{y}_{α}\}_{α=1}^{m}$ are i.i.d. isotropic random vectors of $\mathbb{R}^n$, whose components are not necessarily independent. It was shown in arXiv:0710.1346 that if $m,n\rightarrow \infty$, $m/n\rightarrow c\in \lbrack 0,\infty )$, the Normalized Counting Measures of $\{τ_{α}\}_{α=1}^{m}$ converge weakly and $\{\mathbf{y}_α\}_{α=1}^m$ are \textit{good} (see corresponding definition), then the Normalized Counting Measures of eigenvalues of $M_{n}$ converge weakly in probability to a non-random limit found in \cite{Ma-Pa:67}. In this paper we indicate a subclass of good vectors, which we call \textit{very good} and for which the linear eigenvalue statistics of the corresponding matrices converge in distribution to the Gaussian law, i.e., the Central Limit Theorem is valid. An important example of good vectors, studied in arXiv:0710.1346 are the vectors with log-concave distribution. We discuss the conditions for them, guaranteeing the validity of the Central Limit Theorem for linear eigenvalue statistics of corresponding matrices.

math.PR↗

arXiv2011

Non-Gaussian Limiting Laws for the Entries of Regular Functions of the Wigner Matrices

This paper is a continuation of our paper "Fluctuations of Matrix Elements of Regular Functions of Gaussian Random Matrices", J. Stat. Phys. (134), 147--159 (2009), in which we proved the Central Limit Theorem for the matrix elements of differential functions of the real symmetric random Gaussian matrices (GOE). Here we consider the real symmetric random Wigner matrices having independent (modulo symmetry conditions) but not necessarily Gaussian entries. We show that in this case the matrix elements of sufficiently smooth functions of these random matrices have in general another limiting law which coincides essentially with the probability law of matrix entries.

math.PR↗

arXiv2009

Central limit theorem for linear eigenvalue statistics of random matrices with independent entries

We consider $n\times n$ real symmetric and Hermitian Wigner random matrices $n^{-1/2}W$ with independent (modulo symmetry condition) entries and the (null) sample covariance matrices $n^{-1}X^*X$ with independent entries of $m\times n$ matrix $X$. Assuming first that the 4th cumulant (excess) $κ_4$ of entries of $W$ and $X$ is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as $n\to\infty$, $m\to\infty$, $m/n\to c\in[0,\infty)$ with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class $\mathbf{C}^5$). This is done by using a simple ``interpolation trick'' from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially $\mathbb{C}^5$ test function. Here the variance of statistics contains an additional term proportional to $κ_4$. The proofs of all limit theorems follow essentially the same scheme.

math.PR↗

arXiv2007

On a Random Matrix Models of Quantum Relaxation

Earlier two of us (J.L. and L.P.) considered a matrix model for a two-level system interacting with a $n\times n$ reservoir and assuming that the interaction is modelled by a random matrix. We presented there a formula for the reduced density matrix in the limit $n\to \infty $ as well as several its properties and asymptotic forms in various regimes. In this paper we give the proofs of the assertions, and present also a new fact about the model.

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↗

arXiv2006

Limiting Laws of Linear Eigenvalue Statistics for Unitary Invariant Matrix Models

We study the variance and the Laplace transform of the probability law of linear eigenvalue statistics of unitary invariant Matrix Models of n-dimentional Hermitian matrices as n tends to infinity. Assuming that the test function of statistics is smooth enough and using the asymptotic formulas by Deift et al for orthogonal polynomials with varying weights, we show first that if the support of the Density of States of the model consists of two or more intervals, then in the global regime the variance of statistics is a quasiperiodic function of n generically in the potential, determining the model. We show next that the exponent of the Laplace transform of the probability law is not in general 1/2variance, as it should be if the Central Limit Theorem would be valid, and we find the asymptotic form of the Laplace transform of the probability law in certain cases.

math.PR↗

arXiv2004

Bistability between different dissipative solitons in nonlinear optics

We report the observation of different localized structures coexisting for the same parameter values in an extended system. The experimental findings are carried out in a nonlinear optical interferometer, and are fully confirmed by numerical simulations. The existence of each kind of localized structure is put in relation to a corresponding delocalized pattern observed. Quantitative evaluation of the range of pump parameter allowing bistability between localized structures is given. The dependence of the phenomena on the other relevant parameters is discussed.

nlin.PS↗

arXiv2004

Experimental targeting and control of spatiotemporal chaos in nonlinear optics

We demonstrate targeting and control over spatiotemporal chaos in an optical feedback loop experiment. Different stationary target patterns are stabilized in real-time by means of a two dimensional space extended perturbation field driven by an interfaced computer and applied in real-space to a liquid crystal display device inserted within a control optical loop. The flexibility of the system in switching between different target patterns is also demonstrated.

nlin.CD↗

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↗

arXiv2003

On the Mott formula for the a.c. conductivity and binary correlators in the strong localization regime of disordered systems

We present a method that allows us to find asymptotic form of various characteristics of disordered systems in the strong localization regime, i.e., when either the random potential is big enough or the energy is close enough to the spectrum edges. The method is based on the hypothesis that relevant realizations of the random potential in the strong localization regime have the form of deep random wells that are uniformly and chaotically distributed in the space with a sufficiently small density. Assuming this and using the density expansion, we show first that the density of wells coincides in the leading order with the density of states. Thus the density of states is in fact the small parameter of the theory in the strong localization regime. Then we derive the Mott formula for the low frequency conductivity and the asymptotic formulas for certain two-point correlators when the difference of respective energies is small.

math-ph↗

arXiv2003

Detecting local synchronization in coupled chaotic systems

We introduce a technique to detect and quantify local functional dependencies between coupled chaotic systems. The method estimates the fraction of locally syncronized configurations, in a pair of signals with an arbitrary state of global syncronization. Application to a pair of interacting Rossler oscillators shows that our method is capable to quantify the number of dynamical configurations where a local prediction task is possible, also in absence of global synchronization features.

nlin.CD↗