A Local Limit Theorem and Delocalization of Eigenvectors for Polynomials in Two Matrices

2019/03/19 by Ching-Wei Ho, Ho, Ching-Wei
Mathematics · #Random Matrices and Applications #Advanced Combinatorial Mathematics #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1903.08135

Abstract

We propose a boundary regularity condition for the M_n(ℂ)-valued subordination functions in free probability to prove the local limit theorem and delocalization of eigenvectors for polynomials in two random matrices. We prove this through estimating the pair of M_n(ℂ)-valued approximate subordination functions for the sum of two M_n(ℂ)-valued random matrices γ₁⊗ C_N+γ₂⊗ U_N^*D_NU_N, where C_N, D_N are deterministic diagonal matrices, and U_N is Haar unitary.

Citations

• Limit laws for Random matrices and free products

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