On the distribution of rank of a random matrix over a finite field

Abstract

Let M = (mij) be a random n × n matrix over GF(t) in which each matrix entry mij is independently and identically distributed, with Pr(mij = 0) = 1 − p(n) and Pr(mij = r) = p(n)/(t − 1), r 6= 0. If we choose t ≥ 3, and condition on M having no zero rows or columns, then the probability that M is non-singular tends to ct ∼ ∏∞ j=1(1 − t−j) provided p ≥ (log n + d)/n, where d → −∞ slowly.