Asymptotics in random $(O, 1)$-matrices

Abstract:

Let ${M^n}(i)$ be the class of $n \times n(0,1)$-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 ${M^n}(i)$. We succeed for $i > {n^{3/2 + \varepsilon }}$ 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 ${M^n}(i)$ 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.