The law of large numbers for permanents of random stochastic matrices

A. N. Timashev

Diskretnaya Matematika, Tome 11 (1999) no. 3, pp. 91-98.

Résumé

We consider the class of all $n\times n$ $(0,1)$-matrices with $r$ ones in each row, $2\le r\le n$. For a matrix $P$ chosen randomly and equiprobably from this class, we present sufficient conditions under which the law of large numbers for the permanent $\operatorname{per}P$ is valid in the triangular array scheme as $n\to\infty$ and the parameter $r=r(n)\to\infty$ so that $\sqrt{n}=o(r)$. A similar problem is solved for random $n\times n$ stochastic matrices whose rows are independent $n$-dimensional random variables which are identically distributed by the Dirichlet law with parameter $\nu$ under the condition that $n\to\infty$ and the parameter $\nu=\nu(n)>0$ varies so that $n\nu^2\to\infty$.