Random integral matrices: universality of surjectivity and the cokernel

TL;DR

This paper proves that the probability distributions of cokernels and surjectivity of large random integer matrices are universal and follow Cohen-Lenstra heuristics, regardless of entry distribution, including sparse and graph Laplacian matrices.

Contribution

It establishes asymptotic universality of cokernel distributions for broad classes of random matrices, extending Cohen-Lenstra heuristics to sparse and graph Laplacian matrices.

Findings

Cokernel distributions follow precise formulas involving zeta values.

Surjectivity probability converges to a universal limit.

Results apply to sparse matrices and Laplacians of random digraphs.

Abstract

For a random matrix of entries sampled independently from a fairly general distribution in Z we study the probability that the cokernel is isomorphic to a given finite abelian group, or when it is cyclic. This includes the probability that the linear map between the integer lattices given by the matrix is surjective. We show that these statistics are asymptotically universal (as the size of the matrix goes to infinity), given by precise formulas involving zeta values, and agree with distributions defined by Cohen and Lenstra, even when the distribution of matrix entries is very distorted. Our method is robust and works for Laplacians of random digraphs and sparse matrices with the probability of an entry non-zero only n^{-1+epsilon}.