# 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 ## References SHOWING 1-10 OF 35 REFERENCES • Mathematics • 2014 We apply the general theory of Cauchy biorthogonal polynomials developed in Bertola et al. (Commun Math Phys 287(3):983–1014, 2009) and Bertola et al. (J Approx Th 162(4):832–867, 2010) to the case • Mathematics • 2008 We introduce a new class of two(multi)-matrix models of positive Hermitian matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more • Mathematics • 2012 We consider the product of n complex non-Hermitian, independent random matrices, each of size N × N with independent identically distributed Gaussian entries (Ginibre matrices). The joint probability • Mathematics • 2016 The squared singular values of the product of M complex Ginibre matrices form a biorthogonal ensemble, and thus their distribution is fully determined by a correlation kernel. The kernel permits a • Mathematics • 2014 We investigate the spectral properties of the product of M complex non-Hermitian random matrices that are obtained by removing L rows and columns of larger unitary random matrices uniformly The product of M complex random Gaussian matrices of size N has recently been studied by Akemann, Kieburg, and Wei. They showed that, for fixed M and N, the joint probability distribution for the • Mathematics • 2014 Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random N × N matrices, also called the Ginibre ensemble, we rederive the We consider the gap probability for the Generalized Bessel process in the single-time and multi-time case, a determinantal process which arises as critical limiting kernel in the study of • Mathematics • 2008 The eigenvalue statistics of a pair (M1, M2) of n × n Hermitian matrices taken randomly with respect to the measure$\${1 \over Z_{n}} \exp \left({-n} {\rm Tr}\,(V(M_{1})+ W(M_{2})- \tau
• Mathematics
• 1999
We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with