Spectral statistics of interpolating random circulant matrix and its applications to random circulant graphs

TL;DR

This paper analyzes the spectral properties of a flexible random circulant matrix model, deriving exact joint densities, exploring eigenvalue distribution interpolation, and validating results through simulations and graph spectra comparisons.

Contribution

It introduces a comprehensive spectral analysis of a general interpolating random circulant matrix model, including exact joint densities and eigenvalue distribution interpolation.

Findings

Exact joint probability density function derived as multivariate Gaussian.

Eigenvalue distributions can be smoothly interpolated by adjusting Gaussian parameters.

Monte Carlo simulations and graph spectra confirm analytical results.

Abstract

We consider a versatile matrix model of the form ${\bf A}+i {\bf B}$, where ${\bf A}$ and ${\bf B}$ are real random circulant matrices with independent but, in general, nonidentically distributed Gaussian entries. For this model, we derive exact results for the joint probability density function and find that it is a multivariate Gaussian. Arbitrary order marginal density therefore also readily follows. It is demonstrated that by adjusting the averages and variances of the Gaussian elements of ${\bf A}$ and ${\bf B}$, we can interpolate between a remarkably wide range of eigenvalue distributions in the complex plane. In particular, we can examine the crossover between a random real circulant matrix and a random complex circulant matrix. We also extend our study to include Wigner-like and Wishart-like matrices constructed from our general random circulant matrix. To validate our analytical findings, Monte Carlo simulations are conducted, which confirm the accuracy of our results. Additionally, we compare our analytical results with the spectra of adjacency matrices from various random circulant graphs. Despite the difference in entry distributions-Gaussian in our model and non-Gaussian in the adjacency matrices-the densities show excellent agreement in the large-dimension limit.