Hitting time theorems for random matrices

TL;DR

This paper investigates the hitting time for invertibility in random matrices, showing that invertibility typically occurs when the last zero row or column disappears, extending previous results with quantitative bounds.

Contribution

It establishes new hitting time theorems for invertibility in random matrices, including symmetric Bernoulli matrices, with precise probabilistic bounds.

Findings

Invertibility coincides with the disappearance of zero rows or columns

Results hold with high probability as matrix size grows

Provides quantitative bounds for related random matrix problems

Abstract

Starting from an n-by-n matrix of zeros, choose uniformly random zero entries and change them to ones, one-at-a-time, until the matrix becomes invertible. We show that with probability tending to one as n tends to infinity, this occurs at the very moment the last zero row or zero column disappears. We prove a related result for random symmetric Bernoulli matrices, and give quantitative bounds for some related problems. These results extend earlier work by Costello and Vu [arXiv:math/0606414].