Related papers: Symmetric Function Theory and Unitary Invariant En…
The paper investigates the problem of performing correlation analysis when the number of observations is very large. In such a case, it is often necessary to combine the random observations to achieve dimensionality reduction of the…
Exact evaluation of $<{\rm Tr} S^p>$ is here performed for real symmetric matrices $S$ of arbitrary order $n$, up to some integer $p$, where the matrix entries are independent identically distributed random variables, with an arbitrary…
The lowest representatives of the Form Factors relative to the trace operators of N=1 Super Sinh-Gordon Model are exactly calculated. The novelty of their determination consists in solving a coupled set of unitarity and crossing equations.…
In this paper, we pursue our study of asymptotic properties of families of random matrices that have a tensor structure. In previous work, the first- and second-named authors provided conditions under which tensor products of unitary random…
We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product…
The relation between the Seiberg-Witten prepotentials, Nekrasov functions and matrix models is discussed. We derive quasiclassically the matrix models of Eguchi-Yang type, describing the instantonic contribution to the deformed partition…
A new class of Random Matrix Ensembles is introduced. The Gaussian orthogonal, unitary, and symplectic ensembles GOE, GUE, and GSE, of random matrices are analogous to the classical Gibbs ensemble governed by Boltzmann's distribution in the…
We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble,…
We show that as $n$ changes, the characteristic polynomial of the $n\times n$ random matrix with i.i.d. complex Gaussian entries can be described recursively through a process analogous to P\'olya's urn scheme. As a result, we get a random…
There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size $N$…
We study square integrable functions on the metaplectic group and functions on the space of unitary symmetric matrices. We relate them using the oscillator representations.
We take advantage of the combinatorial interpretations of many sequences of polynomials of binomial type to define a sequence of symmetric functions corresponding to each sequence of polynomials of binomial type. We derive many of the…
Riemannian Gaussian distributions were initially introduced as basic building blocks for learning models which aim to capture the intrinsic structure of statistical populations of positive-definite matrices (here called covariance…
We link the appearance of universal kernels in random matrix ensembles to the phenomenon of shock formation in some fluid dynamical equations. Such equations are derived from Dyson's random walks after a proper rescaling of the time. In the…
A connection between representation of compact groups and some invariant ensembles of Hermitian matrices is described. We focus on two types of invariant ensembles which extend the Gaussian and the Laguerre Unitary ensembles. We study them…
An attempt is made to describe random matrix ensembles with unitary invariance of measure (UE) in a unified way, using a combination of Tracy-Widom (TW) and Adler-Shiota-Van Moerbeke (ASvM) approaches to derivation of partial differential…
We apply symmetric function theory to study random processes formed by singular values of products of truncations of Haar distributed symplectic and orthogonal matrices. These product matrix processes are degenerations of Macdonald…
In his classic book on symmetric functions, Macdonald describes a remarkable result by Green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions. This connection…
The leading correction to the smoothed connected energy density-density correlation function is obtained for the large energy difference, within the context of the Gaussian Random Matrix Theory. In order to achieve this result, the…
Equilibrium Statistical Mechanics is undoubtedly a cornerstone for the description of many particle systems. The common interpretation is based on ensemble theory as put forward by Gibbs, alongside the basic assumptions that different…