The rank of sparse symmetric matrices over arbitrary fields

TL;DR

This paper investigates the asymptotic normalized rank of sparse symmetric matrices over arbitrary fields, showing it converges to a field-independent constant, extending previous results for non-weighted Erdős-Rényi matrices.

Contribution

It extends the understanding of the rank behavior of sparse symmetric matrices over any field, generalizing prior work limited to real fields and non-weighted matrices.

Findings

Normalized rank converges in probability to a constant.

The limiting constant is independent of edge weights and the field.

The result generalizes previous findings for non-weighted Erdős-Rényi matrices.

Abstract

Let $\FF$ be an arbitrary field and $(\bm{G}_{n,d/n})_n$ be a sequence of sparse weighted Erd\H{o}s-R\'enyi random graphs on $n$ vertices with edge probability $d/n$, where weights from $\FF \setminus\{0\}$ are assigned to the edges according to a fixed matrix $J_n$. We show that the normalised rank of the adjacency matrix of $(\bm{G}_{n,d/n})_n$ converges in probability to a constant, and derive the limiting expression. Our result shows that for the general class of sparse symmetric matrices under consideration, the asymptotics of the normalised rank are independent of the edge weights and even the field, in the sense that the limiting constant for the general case coincides with the one previously established for adjacency matrices of sparse (non-weighted) Erd\H{o}s-R\'enyi matrices over $\RR$ from \cite{bordenave2011rank}. Our proof, which is purely combinatorial in its nature, is based on an intricate extension of the novel perturbation approach from \cite{coja2022rank} to the symmetric setting.