On the probability that a random ±1-matrix is singular

Abstract

We report some progress on the old problem of estimating the probability, P n {P_n} , that a random n × n ± 1 n \times n \pm 1 -matrix is singular: Theorem . There is a positive constant ε \varepsilon for which P n > ( 1 − ε ) n {P_n} > {(1 - \varepsilon )^n} . This is a considerable improvement on the best previous bound, P n = O ( 1 / n ) {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 {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 a _ ∈ R n \underline a \in {{\mathbf {R}}^n} is orthogonal to a random ε _ ∈ { ± 1 } n \underline \varepsilon \in {\{ \pm 1\} ^n} to the corresponding probability when ε _ \underline \varepsilon is random from { − 1 , 0 , 1 } n {\{ - 1,0,1\} ^n} with P r ( ε i = − 1 ) = P r ( ε i = 1 ) = p Pr({\varepsilon _i} = - 1) = Pr({\varepsilon _i} = 1) = p and ε i {\varepsilon _i} ’s chosen independently.