Large Deviations for a Non-Centered Wishart Matrix

TL;DR

This paper establishes a large deviation principle for the spectral measures of a perturbed Wishart matrix, linking it to a vector equilibrium problem involving logarithmic energies and Coulomb gas representations.

Contribution

It introduces a novel large deviation analysis for non-centered Wishart matrices using a Coulomb gas approach and a vector equilibrium framework.

Findings

Large deviation principle proven for spectral measures.

Connection established between spectral distribution and vector equilibrium problem.

New Coulomb gas representation provides insight into spectral distribution limits.

Abstract

We investigate an additive perturbation of a complex Wishart random matrix and prove that a large deviation principle holds for the spectral measures. The rate function is associated to a vector equilibrium problem coming from logarithmic potential theory, which in our case is a quadratic map involving the logarithmic energies, or Voiculescu's entropies, of two measures in the presence of an external field and an upper constraint. The proof is based on a two type particles Coulomb gas representation for the eigenvalue distribution, which gives a new insight on why such variational problems should describe the limiting spectral distribution. This representation is available because of a Nikishin structure satisfied by the weights of the multiple orthogonal polynomials hidden in the background.