The singularity probability of a random symmetric matrix is exponentially small

Abstract

Let A A be drawn uniformly at random from the set of all n × n n\times n symmetric matrices with entries in { − 1 , 1 } \{-1,1\} . We show that \[ P ( det ( A ) = 0 ) ⩽ e − c n , \mathbb {P}( \det (A) = 0 ) \leqslant e^{-cn}, \] where c > 0 c>0 is an absolute constant, thereby resolving a long-standing conjecture.