# Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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## On the probability that a random $\pm 1$-matrix is singularHTML articles powered by AMS MathViewer

by Jeff Kahn, János Komlós and Endre Szemerédi
J. Amer. Math. Soc. 8 (1995), 223-240
DOI: https://doi.org/10.1090/S0894-0347-1995-1260107-2

## Abstract:

We report some progress on the old problem of estimating the probability, ${P_n}$, that a random $n \times n \pm 1$-matrix is singular: Theorem. There is a positive constant $\varepsilon$ for which ${P_n} < {(1 - \varepsilon )^n}$. This is a considerable improvement on the best previous bound, ${P_n} = O(1/\sqrt n )$, given by Komlós in 1977, but still falls short of the often-conjectured asymptotical formula ${P_n} = (1 + o(1)){n^2}{2^{1 - n}}$. The proof combines ideas from combinatorial number theory, Fourier analysis and combinatorics, and some probabilistic constructions. A key ingredient, based on a Fourier-analytic idea of Halász, is an inequality (Theorem 2) relating the probability that $\underline a \in {{\mathbf {R}}^n}$ is orthogonal to a random $\underline \varepsilon \in {\{ \pm 1\} ^n}$ to the corresponding probability when $\underline \varepsilon$ is random from ${\{ - 1,0,1\} ^n}$ with $Pr({\varepsilon _i} = - 1) = Pr({\varepsilon _i} = 1) = p$ and ${\varepsilon _i}$’s chosen independently.
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