Global eigenvalue fluctuations of random biregular bipartite graphs

TL;DR

This paper analyzes the eigenvalue fluctuations of random biregular bipartite graphs, providing new bounds and laws that enhance understanding of their spectral properties and applications to hypergraphs.

Contribution

It introduces a total variation bound for cycle counts and establishes a semicircle law for certain bipartite graphs, advancing spectral analysis techniques.

Findings

Eigenvalue fluctuations characterized for large classes of functions.

Total variation bound for cycle and walk counts in biregular bipartite graphs.

Semicircle law proven for graphs with diverging degree ratios.

Abstract

We compute the eigenvalue fluctuations of uniformly distributed random biregular bipartite graphs with fixed and growing degrees for a large class of analytic functions. As a key step in the proof, we obtain a total variation distance bound for the Poisson approximation of the number of cycles and cyclically non-backtracking walks in random biregular bipartite graphs, which might be of independent interest. We also prove a semicircle law for random $(d_1,d_2)$-biregular bipartite graphs when $\frac{d_1}{d_2}\to\infty$. As an application, we translate the results to adjacency matrices of uniformly distributed random regular hypergraphs.