Asymptotics in random (0, 1)-matrices

Abstract

Let Mn(i) be the class of n × n (0, l)-matrices with i ones. We wish to find the first and second moments of Perm B, the permanent of the matrix B, as B ranges over the class Mn(i). We succeed for i>n3/2+ in finding an asymptotic estimate of these quantities. It turns out that the square of the first moment is asymptotic to the second moment, so we may conclude that almost all matrices in Mn(i) have asymptotically the same permanent. It is suggested that the technique employed will also enable us to evaluate asymptotically the number of hamiltonian circuits in a random graph with i links on n vertices. © 1970 American Mathematical Society.