On the singularity probability of random Bernoulli matrices

Tao, Terence; Vu, Van

doi:10.1090/S0894-0347-07-00555-3

On the singularity probability of random Bernoulli matrices
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by Terence Tao and Van Vu

J. Amer. Math. Soc. 20 (2007), 603-628

DOI: https://doi.org/10.1090/S0894-0347-07-00555-3

Published electronically: February 6, 2007

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Abstract:

Let $n$ be a large integer and let $M_n$ be a random $n \times n$ matrix whose entries are i.i.d. Bernoulli random variables (each entry is $\pm 1$ with probability $1/2$). We show that the probability that $M_n$ is singular is at most $(3/4 +o(1))^n$, improving an earlier estimate of Kahn, Komlós and Szemerédi, as well as earlier work by the authors. The key new ingredient is the applications of Freiman-type inverse theorems and other tools from additive combinatorics.

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