Local and global universality of random matrix cokernels

TL;DR

This paper investigates the universal behavior of cokernels in various random matrix models, establishing broad conditions for universality and solving a longstanding question about sandpile groups of random graphs.

Contribution

It introduces a systematic method to prove universality of cokernel distributions across diverse random matrix models, including symmetric, skew-symmetric, and Laplacian matrices.

Findings

Proves local and global universality of cokernel statistics

Determines the probability that a sandpile group of an Erdős-Rényi graph is cyclic

Answers a question posed by Lorenzini in 2008

Abstract

In this paper we study the cokernels of various random integral matrix models, including random symmetric, random skew-symmetric, and random Laplacian matrices. We provide a systematic method to establish universality under very general randomness assumption. Our highlights include both local and global universality of the cokernel statistics of all these models. In particular, we find the probability that a sandpile group of an Erdos-Renyi random graph is cyclic, answering a question of Lorenzini from 2008.