The characteristic polynomial of sums of random permutations and regular digraphs

TL;DR

This paper studies the spectral properties of sums of random permutation matrices and regular digraphs, showing convergence of their characteristic polynomials and providing an elementary proof of spectral gaps.

Contribution

It introduces a new approach to analyze the characteristic polynomial of sums of random permutation matrices and applies it to prove spectral gaps in random regular digraphs.

Findings

Normalized characteristic polynomial converges to a random analytic function

Elementary proof of spectral gap in random regular digraphs

Results hold for fixed and slowly growing degree d

Abstract

Let $A_n$ be the sum of $d$ permutation matrices of size $n\times n$, each drawn uniformly at random and independently. We prove that the normalized characteristic polynomial $\frac{1}{\sqrt{d}}\det(I_n - z A_n/\sqrt{d})$ converges when $n\to \infty$ towards a random analytic function on the unit disk. As an application, we obtain an elementary proof of the spectral gap of random regular digraphs. Our results are valid both in the regime where $d$ is fixed and for $d$ slowly growing with $n$.