On the singularity probability of random Bernoulli matrices

Details On the singularity probability of random Bernoulli matrices On the singularity probability of random Bernoulli matrices

2005-01-20

arxiv.org/abs/math/0501313v2

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

Mathematics

Combinatorics

30 pages, no figures. This is the final version

Scientific paper

Let $n$ be a large integer and $M_n$ be a random $n$ by $n$ 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, Koml\'os and Szemer\'edi, 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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