Limit distribution of the probabilities of the permanent of a random matrix in the field $\operatorname{GF}(p)$

We prove that the permanent $\per(A_{nm})$ of a random $n\times m$ matrix $A_{nm}$ with elements from $\GF(p)$ and independent rows has the limit distribution of the form \[ p_k = \lim_{n\to\infty} \P\{\per(A_{nm}) = k\} = \rho_m\delta_{k0} + (1-\rho_m)/p, \qquad k=0,1,2,\ldots,p-1, \] where $\delta_{k0}$ is Kronecker's symbol. This distribution for each $m$ coincides with the probability distribution of some function of independent random variables uniformly distributed on $\GF(p)$.
This work was supported by the Russian Foundation of Basic Research, Grant 93–011–1443.