Chio Condensation and Random Sign Matrices

Abstract

This is to suggest a new approach to the old and open problem of counting the number f_n of Z-singular n x n matrices with entries from {-1,+1}: Comparison of two measures, none of them the uniform measure, one of them closely related to it, the other asymptotically under control by a recent theorem of Bourgain, Vu and Wood. We will define a measure P_chio on the set {-1,0,+1}^([n-1]^2) of all (n-1)x(n-1)-matrices with entries from {-1,0,+1} which (owing to a determinant identity published by M. F. Chio in 1853) is closely related to the uniform measures on {-1,+1}^([n]^2) and {0,1}^([n-1]^2) and at the same time it intriguingly mimics the so-called lazy coin flip distribution P_lcf on {-1,0,+1}^([n-1]^2), with the resemblance fading more and more as the events get smaller. This is relevant in view of a recent theorem of J. Bourgain, V. H. Vu and P. M. Wood (J. Funct. Anal. 258 (2010), 559--603) which proves that if the entries of an n x n matrix whose {-1,0,+1}-entries are governed by P_lcf and fully independent (they are not when governed by P_chio), then an asymptotically optimal bound on the singularity probability over Z can be proved. We will characterize P_chio graph-theoretically and use the characterization to prove that given a B in {-1,0,+1}^([n-1]^2), deciding whether P_chio[B] = P_lcf[B] is equivalent to deciding an evasive graph property, hence the time complexity of this decision is Omega(n^2). Moreover, we will prove k-wise independence properties of P_chio. Many questions suggest themselves that call for further work. In particular, the present paper will close with more constrained equivalent formulations of the conjecture f_n/2^(n^2) ~ (1/2 + o(1))^n.