On the Universality of the Non-singularity of General Ginibre and Wigner Random Matrices

TL;DR

This paper proves that large classes of random matrices, including Ginibre and Wigner ensembles with diverse entry distributions, are almost surely non-singular, with universal convergence rates and no moment restrictions.

Contribution

It establishes the universal non-singularity of general Ginibre and Wigner matrices under minimal assumptions, extending previous results to broader models including sparse and banded matrices.

Findings

Almost sure non-singularity of general Ginibre and Wigner matrices.

Universal convergence rates depending only on the largest jump of entry distributions.

No moment assumptions required for the distributions of matrix entries.

Abstract

We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on the size of the matrix. These models include adjacency matrices of random graphs and also sparse, generalized, universal and banded random matrices. We find universal rates of convergence and precise estimates for the probability of singularity which depend only on the size of the biggest jump of the distribution functions governing the entries of the matrix and not on the range of values of the random entries. Moreover, no moment assumptions are made about the distributions governing the entries. Our proofs are based on a concentration function inequality due to Kolmogorov, Rogozin and Kesten, which allows us to improve universal rates of convergence for the Wigner case when the distribution of the entries do not depend on the size of the matrix.