Some results on random circulant matrices

TL;DR

This paper investigates the spectral properties and singularity probabilities of random circulant matrices, revealing their spectral distribution as complex normal and providing bounds on their singularity likelihood.

Contribution

It establishes the limiting spectral distribution for random circulant matrices and bounds the probability of their singularity, extending results from non-Hermitian random matrix theory.

Findings

Spectral distribution of random circulant matrices is complex normal.

Bounds are provided for the probability that a circulant sign matrix is singular.

Results are analogous to recent theorems on non-Hermitian random matrices.

Abstract

This paper considers random (non-Hermitian) circulant matrices, and proves several results analogous to recent theorems on non-Hermitian random matrices with independent entries. In particular, the limiting spectral distribution of a random circulant matrix is shown to be complex normal, and bounds are given for the probability that a circulant sign matrix is singular.