Wigner type laws for structured random matrices

TL;DR

This paper investigates the spectral distribution of large structured random matrices with patterned entries, establishing asymptotic moment convergence and conditions for asymptotic freeness, extending classical Wigner matrix results.

Contribution

It introduces a framework for analyzing eigenvalue distributions of patterned matrices and proves asymptotic moment convergence and freeness under certain conditions.

Findings

Eigenvalue distributions follow a specific limiting distribution.

Moments of matrices converge to integrals over combinatorial path counts.

Patterned matrices can be asymptotically free under certain conditions.

Abstract

For a sufficiently nice 2 dimensional shape, we define its approximating matrix (or patterned matrix) as a random matrix with iid entries arranged according to a given pattern. For large approximating matrices, we observe that the eigenvalues roughly follow an underlying distribution. This phenomenon is similar to the classical observation on Wigner matrices. We prove that the moments of such matrices converge asymptotically as the size increases and equals to the integral of a combinatorially-defined function which counts certain paths on a finite grid. We also consider the case of several independent patterned matrices. Under a specific set of conditions, these matrices admit asymptotic freeness with respect to full-filled independent square random matrices. In our conclusion, we present several open problems.