Asymptotic Expansion of Spherical Integral
Jiaoyang Huang

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.
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Taxonomy
TopicsRandom Matrices and Applications · Spectral Theory in Mathematical Physics · Advanced Algebra and Geometry
