Cycles and eigenvalues of sequentially growing random regular graphs

TL;DR

This paper studies the eigenvalue fluctuations of growing random regular graphs, revealing they can be described by independent Yule processes and exhibit GFF-like properties as the degree increases.

Contribution

It introduces a novel coupling of random regular graphs via the Chinese Restaurant Process and characterizes eigenvalue fluctuations using Yule processes.

Findings

Eigenvalue statistics relate to independent Yule processes.

As degree grows, GFF-like properties emerge.

Eigenvalue fluctuations differ from Wigner matrices, showing Poisson behavior.

Abstract

Consider the sum of $d$ many i.i.d. random permutation matrices on $n$ labels along with their transposes. The resulting matrix is the adjacency matrix of a random regular (multi)-graph of degree $2d$ on $n$ vertices. It is known that the distribution of smooth linear eigenvalue statistics of this matrix is given asymptotically by sums of Poisson random variables. This is in contrast with Gaussian fluctuation of similar quantities in the case of Wigner matrices. It is also known that for Wigner matrices the joint fluctuation of linear eigenvalue statistics across minors of growing sizes can be expressed in terms of the Gaussian Free Field (GFF). In this article, we explore joint asymptotic (in $n$) fluctuation for a coupling of all random regular graphs of various degrees obtained by growing each component permutation according to the Chinese Restaurant Process. Our primary result is that the corresponding eigenvalue statistics can be expressed in terms of a family of independent Yule processes with immigration. These processes track the evolution of short cycles in the graph. If we now take $d$ to infinity, certain GFF-like properties emerge.