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Leonid Pastur

Publications and source records attributed to Leonid Pastur.

18 recordsLinked to original sources
arXiv2025

Stochastic gyration driven by dichotomous noises

We consider stochastic dynamics of a particle on a plane in presence of two noises and a confining parabolic potential - an analog of the experimentally-relevant Brownian Gyrator (BG) model. In contrast to the standard BG model, we suppose here that the time-evolution of the position components is driven not by Gaussian white-noises, but by two statistically-independent dichotomous noises. We calculate analytically the position variances and cross-correlations, as well as the mean angular momentum, which permits us to establish the conditions in which a spontaneous rotational motion of the particle around the origin takes place. We also present a numerical analysis of the mean angular velocity. Lastly, we calculate analytically some marginal position probability density functions revealing a remarkably rich behavior that emerges in such a system of two coupled linear stochastic differential equations. We show that depending on the values of parameters characterizing noises these distributions approach the steady-state forms defined on a finite support, having very unusual shapes, possessing multiple maxima and minima, plateaus and exhibiting a discontinuous behavior.

cond-mat.stat-mech↗
arXiv2025

Area Law for the entanglement entropy of free fermions in nonrandom ergodic field

This paper deals with the asymptotic behaviour of a widely used correlation characteristic in large quantum systems. The correlations are known as quantum entanglement, the characteristic is called entanglement entropy, and the system is an ideal gas of spinless lattice fermions. The system is determined by its one-body Hamiltonian. It is shown in EPS [18] that if the Hamiltonian is an ergodic finite difference operator with an exponentially decaying spectral projection, then the asymptotic form of the entanglement entropy is the so-called area law. However, the only class of one-body Hamiltonians for which this spectral condition was verified is that consisting of discrete Schr\"odinger operators with random potential. In this paper, we prove the validity of the area law for several classes of Schr\"odinger operators whose potentials are ergodic but not random. We begin with quasiperiodic and limit-periodic operators and then move on to the interesting and highly non-trivial case of potentials generated by subshifts of finite type. These arose in the theory of dynamical systems when studying non-random chaotic phenomena. The corresponding asymptotic study requires quite an involved spectral analysis. Consequently, the majority of the paper is devoted to the proof and application of a variety of spectral properties of the operators in question, in particular we prove uniform localisation of the ejgenfunctions for the Maryland model and the exponential decay of the eihgenfunction correlator for a variety of models . We believe that these properties are of considerable independent interest.

math-ph↗
arXiv2025

Ergodic Hankel operators

We introduce a new class of operators: ergodic families of self-adjoint Hankel operators realised as integral operators on the half-line. Inspired by the spectral theory of differential and finite-difference operators with ergodic coefficients, we develop a spectral theory of ergodic Hankel operators. For these operators, we define the Integrated Density of States (IDS) measure and establish its fundamental properties. In particular, we determine the total mass of the IDS measure in the positive semi-definite case. We also consider in more detail two classes of ergodic Hankel operators for which we are able to make further progress: periodic Hankel operators and the random Kronig--Penney--Hankel (rKPH) model. For periodic Hankel operators, we prove that the IDS measure is a sum of a pure point and absolutely continuous components, and describe the structure of both components. For the rKPH model, we prove the counterparts of the cornerstone results of the spectral theory of random Schr\"odinger operators: Lifshitz tails at the edges of the spectrum, the Wegner bound and Anderson localisation in a natural asymptotic regime. We conclude with some open problems.

math.SP↗
arXiv2025

Sums of projections with random coefficients

We study infinite sums \[ {\mathcal P}_{\varkappa}=\sum_{n=-\infty}^\infty \varkappa_n \langle\cdot, \psi_n\rangle\psi_n \] of rank-one projections in a Hilbert space, where $\{\psi_n\}_{n\in\mathbb Z}$ are norm-one vectors, not necessarily orthogonal, and $\{\varkappa_n\}_{n\in\mathbb Z}$ are independent identically distributed positive random variables. Assuming that the Gram matrix $\{\langle\psi_n,\psi_m\rangle\}_{n,m\in\mathbb Z}$ defines a bounded operator on $\ell^2(\mathbb Z)$ and that its entries depend only on the difference $n-m$, we analyse ${\mathcal P}_{\varkappa}$ within the framework of spectral theory of ergodic operators. Inspired by the spectral theory of ergodic Schr\"odinger operators, we define the integrated density of states (IDS) measure $\nu_{{\mathcal P}_\varkappa}$ for ${\mathcal P}_{\varkappa}$ and establish results on its continuity and absolute continuity, including Wegner-type estimates and Lifshitz tail behaviour near the spectral edges. In the asymptotic regime of nearly-orthogonal $\psi_n$, we prove the Anderson-type localisation result: the spectrum of ${\mathcal P}_{\varkappa}$ is pure point almost surely.

math.SP↗
arXiv2023

Irregular gyration of a two-dimensional random-acceleration process in a confining potential

We study the stochastic dynamics of a two-dimensional particle assuming that the components of its position are two coupled random-acceleration processes evolving in a confining parabolic potential and are the subjects of independent Gaussian white noises with different amplitudes (temperatures). We determine the standard characteristic properties, i.e., the moments of position's components and their velocities, mixed moments and two-time correlations, as well as the position-velocity probability density function (pdf). We show that if the amplitudes of the noises are not equal, then the particle experiences a non-zero (on average) torque, such that the angular momentum L and the angular velocity W have non-zero mean values. Both are (irregularly) oscillating with time t, such that the characteristics of a rotational motion are changing their signs. We also evaluate the pdf-s of L and W and show that the former has exponential tails for any fixed t, and hence, all moments. In addition, in the large-time limit this pdf converges to a uniform distribution with a diverging variance. The pdf of W possesses heavy power-law tails such that the mean W is the only existing moment. This pdf converges to a limiting form which, surprisingly, is completely independent of the amplitudes of noises.

cond-mat.stat-mech↗
arXiv2022

Eigenvalue Distribution of Large Random Matrices Arising in Deep Neural Networks: Orthogonal Case

The paper deals with the distribution of singular values of the input-output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent rectangular weight matrices alternate with diagonal matrices whose entries depend on the corresponding column of the nearest neighbor weight matrix. The problem was considered in \cite{Pe-Co:18} for the Gaussian weights and biases and also for the weights that are Haar distributed orthogonal matrices and Gaussian biases. Basing on a free probability argument, it was claimed that in these cases the singular value distribution of the Jacobian in the limit of infinite width (matrix size) coincides with that of the analog of the Jacobian with special random but weight independent diagonal matrices, the case well known in random matrix theory. The claim was rigorously proved in \cite{Pa-Sl:21} for a quite general class of weights and biases with i.i.d. (including Gaussian) entries by using a version of the techniques of random matrix theory. In this paper we use another version of the techniques to justify the claim for random Haar distributed weight matrices and Gaussian biases.

stat.ML↗
arXiv2020

The Dynamics of Quantum Correlations of Two Qubits in a Common Environment Modeled by Large Random Matrices

This paper is a continuation of our previous paper [8], in which we have studied the dynamics of quantum correlations of two qubits embedded each into its own disordered multiconnected environment. We modeled the environment by random matrices of large size allowing for a possibility to describe meso- and even nanoenvironments. In this paper we also study the dynamics of quantum correlations of two qubits but embedded into a common environment which we also model by random matrices of large size. We obtain the large size limit of the reduced density matrix of two qubits. We then use an analog of the Bogolyubov-van Hove (also known as the Born-Markov) approximation of the theory of open systems and statistical mechanics. The approximation does not imply in general the Markovian evolution in our model but allows for sufficiently detailed analysis both analytical and numerical of the evolution of several widely used quantifiers of quantum correlation, mainly entanglement. We find a number of new patterns of qubits dynamics comparing with the case of independent environments studied in [8] and displaying the role of dynamical (indirect, via the environment) correlations in the enhancing and diversification of qubit evolution. Our results, (announced in [9]), can be viewed as a manifestation of the universality of certain properties of the decoherent qubit evolution which have been found previously in various exact and approximate versions of two-qubit models with macroscopic bosonic environment.

quant-ph↗
arXiv2020

On Random Matrices Arising in Deep Neural Networks. Gaussian Case

The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important difference is that the population covariance matrices, which are assumed to be non-random in the standard setting of statistics and random matrix theory, are now random, moreover, are certain functions of random data matrices. The problem has been considered in recent work [21] by using the techniques of free probability theory. Since, however, free probability theory deals with population matrices which are independent of the data matrices, its applicability in this case requires an additional justification. We present this justification by using a version of the standard techniques of random matrix theory under the assumption that the entries of data matrices are independent Gaussian random variables. In the subsequent paper [18] we extend our results to the case where the entries of data matrices are just independent identically distributed random variables with several finite moments. This, in particular, extends the property of the so-called macroscopic universality on the considered random matrices.

math-ph↗
arXiv2015

The Shannon's mutual information of a multiple antenna time and frequency dependent channel: an ergodic operator approach

Consider a random non-centered multiple antenna radio transmission channel. Assume that the deterministic part of the channel is itself frequency selective, and that the random multipath part is represented by an ergodic stationary vector process. In the Hilbert space $l^2({\mathbb Z})$, one can associate to this channel a random ergodic self-adjoint operator having a so-called Integrated Density of States (IDS). Shannon's mutual information per receive antenna of this channel coincides then with the integral of a $\log$ function with respect to the IDS. In this paper, it is shown that when the numbers of antennas at the transmitter and at the receiver tend to infinity at the same rate, the mutual information per receive antenna tends to a quantity that can be identified and, in fact, is closely related to that obtained within the random matrix approach. This result can be obtained by analyzing the behavior of the Stieltjes transform of the IDS in the regime of the large numbers of antennas.

cs.IT↗
arXiv2015

On a Limiting Distribution of Singular Values of Random Band Matrices

An equation is obtained for the Stieltjes transform of the normalized distribution of singular values of non-symmetric band random matrices in the limit when the band width and rank of the matrix simultaneously tend to infinity. Conditions under which this limit agrees with the quarter-circle law are found. An interesting particular case of lower triangular random matrices is also considered and certain properties of the corresponding limiting singular value distribution are given.

math-ph↗
arXiv2015

On the Equilibrium State of a Small System with Random Matrix Coupling to Its Environment

We consider a random matrix model of interaction between a small $n$-level system, $S$, and its environment, a $N$-level heat reservoir, $R$. The interaction between $S$ and $R$ is modeled by a tensor product of a fixed $% n\times n$ matrix and a $N\times N$ hermitian Gaussian random matrix. We show that under certain "macroscopicity" conditions on $R$, the reduced density matrix of the system $ρ_{S}=\mathrm{Tr}_{R}ρ_{S\cup R}^{(eq)} $, is given by $ρ_{S}^{(c)}\sim \exp {\{-βH_{S}\}}$, where $H_{S}$ is the Hamiltonian of the isolated system. This holds for all strengths of the interaction and thus gives some justification for using $% ρ_{S}^{(c)}$ to describe some nano-systems, like biopolymers, in equilibrium with their environment \cite{Se:12}. Our results extend those obtained previously in \cite{Le-Pa:03,Le-Co:07} for a special two-level system.

math-ph↗
arXiv2013

Thomas precession, persistent spin currents and quantum forces

We consider T-invariant spin currents induced by spin-orbit interactions which originate from the confined motion of spin carriers in nanostructures. The resulting Thomas spin precession is a fundamental and purely kinematic relativistic effect occurring when the acceleration of carriers is not parallel to their velocity. In the case, where the carriers (e.g. electrons) have magnetic moment the forces due to the electric field of the spin current can, in certain conditions, exceed the van der Waals-Casimir forces by several orders of magnitude. We also discuss a possible experimental set-up tailored to use these forces for checking the existence of a nonzero anomalous magnetic moment of the photon.

cond-mat.mes-hall↗
arXiv2012

Effect of winding edge currents

We discuss persistent currents for particles with internal degrees of freedom. The currents arise because of winding properties essential for the chaotic motion of the particles in a confined geometry. The currents do not change the particle concentrations or thermodynamics, similar to the skipping orbits in a magnetic field.

cond-mat.stat-mech↗
arXiv2007

On the Limiting Empirical Measure of the sum of rank one matrices with log-concave distribution

We consider $n\times n$ real symmetric and hermitian random matrices $H_{n,m}$ equals the sum of a non-random matrix $H_{n}^{(0)}$ matrix and the sum of $m$ rank-one matrices determined by $m$ i.i.d. isotropic random vectors with log-concave probability law and i.i.d. random amplitudes $\{τ_{α}\}_{α=1}^{m}$. This is a generalization of the case of vectors uniformly distributed over the unit sphere, studied in [Marchenko-Pastur (1967)]. We prove that if $n\to \infty, m\to \infty, m/n\to c\in \lbrack 0,\infty)$ and that the empirical eigenvalue measure of $H_{n}^{(0)}$ converges weakly, then the empirical eigenvalue measure of $H_{n,m}$ converges in probability to a non-random limit, found in [Marchenko-Pastur (1967)].

math.PR↗
arXiv2006

A New Approach for Capacity Analysis of Large Dimensional Multi-Antenna Channels

This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to infinity at the same rate. Using a new and simple approach based on Poincaré-Nash inequality and on an integration by parts formula, it is rigorously established that the mutual information converges to a Gaussian random variable whose mean and variance are evaluated. These results confirm previous evaluations based on the powerful but non rigorous replica method. It is believed that the tools that are used in this paper are simple, robust, and of interest for the communications engineering community.

cs.IT↗
arXiv2005

From Random Matrices to Quasiperiodic Jacobi Matrices via Orthogonal Polynomials

We present an informal review of results on asymptotics of orthogonal polynomials, stressing their spectral aspects and similarity in two cases considered. They are polynomials orthonormal on a finite union of disjoint intervals with respect to the Szego weight and polynomials orthonormal on R with respect to varying weights and having the same union of intervals as the set of oscillations of asymptotics. In both cases we construct double infinite Jacobi matrices with generically quasiperiodic coefficients and show that each of them is an isospectral deformation of another. Related results on asymptotic eigenvalue distribution of a class of random matrices of large size are also shortly discussed.

math-ph↗
arXiv1999

A Simple Approach to Global Regime of the Random Matrix Theory

We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles.

math.SP↗