Asymptotic Expansion of Spherical Integral

TL;DR

This paper derives the full asymptotic expansion of spherical integrals involving rank-one matrices and applies these results to establish the asymptotic freeness of Wigner matrices with deterministic Hermitian matrices.

Contribution

It provides the first and second terms of the asymptotic expansion of spherical integrals for rank-one matrices and links these results to asymptotic freeness in random matrix theory.

Findings

Established the existence of full asymptotic expansions for spherical integrals

Derived explicit first and second order terms in the expansion

Demonstrated asymptotic freeness of Wigner and deterministic Hermitian matrices

Abstract

We consider the spherical integral of real symmetric or Hermitian matrices when the rank of one matrix is one. We prove the existence of the full asymptotic expansions of these spherical integrals and derive the first and the second term in the asymptotic expansion. Using asymptotic expression of the spherical integral, we derive the asymptotic freeness of Wigner matrices with (deterministic) Hermitian matrices.