# Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices

@article{Bertola2014UniversalityCA,
title={Universality Conjecture and Results for a Model of Several Coupled Positive-Definite Matrices},
author={Marco Bertola and Thomas Bothner},
journal={Communications in Mathematical Physics},
year={2014},
volume={337},
pages={1077-1141}
}
• Published 9 July 2014
• Mathematics
• Communications in Mathematical Physics
The paper contains two main parts: in the first part, we analyze the general case of $${p \geq 2}$$p≥2 matrices coupled in a chain subject to Cauchy interaction. Similarly to the Itzykson-Zuber interaction model, the eigenvalues of the Cauchy chain form a multi level determinantal point process. We first compute all correlations functions in terms of Cauchy biorthogonal polynomials and locate them as specific entries of a $${(p+1) \times (p+1)}$$(p+1)×(p+1) matrix valued solution of a Riemann…
• Mathematics
• 2015
We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination
• Mathematics
Communications in Mathematical Physics
• 2016
We study the singular values of the product of two coupled rectangular random matrices as a determinantal point process. Each of the two factors is given by a parameter dependent linear combination
• Mathematics
• 2015
In this review we summarise recent results for the complex eigenvalues and singular values of finite products of finite size random matrices, their correlation functions and asymptotic limits. The
• Mathematics
• 2014
It has been shown by Akemann, Ipsen and Kieburg that the squared singular values of products of $M$ rectangular random matrices with independent complex Gaussian entries are distributed according to
• Mathematics
• 2019
Given $K$ arbitrary poles, which are neither necessarily distinct nor bounded, on the extended real line, a corresponding ordered base of rational functions orthogonal with respect to varying
• Mathematics
• 2016
We study the eigenvalue correlations of random Hermitian n×n matrices of the form S=M+ϵH, where H is a GUE matrix, 0\$ ?>ϵ>0, and M is a positive-definite Hermitian random matrix, independent of H,
• Mathematics
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
• 2019
We compare finite rank perturbations of the following three ensembles of complex rectangular random matrices: First, a generalised Wishart ensemble with one random and two fixed correlation matrices
• Mathematics
• 2016
We study the distribution of the smallest eigenvalue for certain classes of positive-definite Hermitian random matrices, in the limit where the size of the matrices becomes large. Their limit
The Muttalib–Borodin ensemble is a probability density function for n particles on the positive real axis that depends on a parameter θ and a weight w. We consider a varying exponential weight that