Asymptotics of k dimensional spherical integrals and applications

TL;DR

This paper establishes that k-dimensional spherical integrals asymptotically behave like products of 1-dimensional integrals, enabling generalizations of large deviations principles for eigenvalues in random matrix theory.

Contribution

It introduces a method to approximate k-dimensional spherical integrals, extending large deviations principles to higher dimensions and more complex matrix models.

Findings

Asymptotic equivalence of k-dimensional and 1-dimensional spherical integrals.

Generalized large deviations principles for eigenvalues of various random matrices.

Proved universality results for large deviations of extreme eigenvalues.

Abstract

In this article, we prove that k-dimensional spherical integrals are asymptotically equivalent to the product of 1-dimensional spherical integrals. This allows us to generalize several large deviations principles in random matrix theory known before only in a one-dimensional case. As examples, we study the universality of the large deviations for k extreme eigenvalues of Wigner matrices (resp. Wishart matrices, resp. matrices with general variance profiles) with sharp sub-Gaussian entries, as well as large deviations principles for extreme eigenvalues of Gaussian Wigner and Wishart matrices with a finite dimensional perturbation.