On the singularity probability of random Bernoulli matrices
Terence Tao, Van Vu
TL;DR
The paper bounds the singularity probability of random Bernoulli matrices by a refined finite-field approach combined with additive combinatorics. A central innovation is a Structure Theorem for exceptional hyperplanes, derived via Freiman-type inverse theorems and Halász-type Fourier analysis, which constrains the defining coordinates to lie in a low-rank generalized arithmetic progression with controlled norm. By partitioning hyperplanes by combinatorial dimension and carefully bounding contributions from small, large, and medium dimensions, the authors achieve the bound $P_n \le (3/4 + o(1))^n$, advancing toward the conjectured $(1/2+o(1))^n$ rate. The work highlights a deep connection between random matrix theory and inverse additive combinatorics, offering a modular framework for tackling singularity probabilities in random matrices.
Abstract
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, 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.