Related papers: Small-Deviation Inequalities for Sums of Random Ma…
This is a brief survey of classical and recent results about the typical behavior of eigenvalues of large random matrices, written for mathematicians and others who study and use matrices but may not be accustomed to thinking about…
Random matrix theory has played an important role in various areas of pure mathematics, mathematical physics, and machine learning. From a practical perspective of data science, input data are usually normalized prior to processing. Thus,…
In this paper, we present a new framework to obtain tail inequalities for sums of random matrices. Compared with existing works, our tail inequalities have the following characteristics: 1) high feasibility--they can be used to study the…
Random-matrix theory is applied to transition-rate matrices in the Pauli master equation. We study the distribution and correlations of eigenvalues, which govern the dynamics of complex stochastic systems. Both the cases of identical and of…
Large random matrices appear in different fields of mathematics and physics such as combinatorics, probability theory, statistics, operator theory, number theory, quantum field theory, string theory etc... In the last ten years, they…
We present an analytical technique to compute the probability of rare events in which the largest eigenvalue of a random matrix is atypically large (i.e.\ the right tail of its large deviations). The results also transfer to the left tail…
This paper is about the relation of random matrix theory and the subordination phenomenon in complex analysis. We find that the resolvent of the sum of two random matrices is approximately subordinated to the resolvents of the original…
The properties of the first (largest) eigenvalue and its eigenvector (first eigenvector) are investigated for large sparse random symmetric matrices that are characterized by bimodal degree distributions. In principle, one should be able to…
We prove deviation inequalities for sums of high-dimensional random matrices and operators with dependence and {\rc heavy tails}. Estimation of high-dimensional matrices is a concern for numerous modern applications. However, most results…
We establish deviation inequalities for the maxima of partial sums of a martingale differences sequence, and of a strictly stationary orthomartingale random field. These inequalities can be used to establish complete convergence of…
The goal of these expository notes is to give an introduction to random matrices for non-specialist of this topic focusing on the link between random matrices and systems of particles in interaction. We first recall some general results…
We introduce a new random matrix model called distance covariance matrix in this paper, whose normalized trace is equivalent to the distance covariance. We first derive a deterministic limit for the eigenvalue distribution of the distance…
We study the large deviations of sums of correlated random variables described by a matrix product ansatz, which generalizes the product structure of independent random variables to matrices whose non-commutativity is the source of…
The goal of this article is to study how much the eigenvalues of large Hermitian random matrices deviate from certain deterministic locations -- or in other words, to investigate optimal rigidity estimates for the eigenvalues. We do this in…
This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the…
Using the diagrammatic method, we derive a set of self-consistent equations that describe eigenvalue distributions of large correlated asymmetric random matrices. The matrix elements can have different variances and be correlated with each…
We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit…
Random matrices now play a role in many parts of computational mathematics. To advance these applications, it is desirable to have tools that are flexible, easy to use, and powerful. Over the last 25 years, researchers have developed a…
We prove that few largest (and most important) eigenvalues of random symmetric matrices of various kinds are very strongly concentrated. This strong concentration enables us to compute the means of these eigenvalues with high precision. Our…
We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent away from the spectral edges, generalizing the…