Asymptotic Behavior of the Expectation Value of Permanent Products, a Sequel

Abstract

Continuing the computations of the previous paper,[1], we calculate another approximation to the expectation value of the product of two permanents in the ensemble of 0-1 n x n matrices with like row and column sums equal r uniformly weighted. Here we consider the Bernoulli random matrix ensemble where each entry independently has a probability p=r/n of being one, otherwise zero. We denote the expectations of the approximation ensemble of [1] by E, and the expectations of the present approximation ensemble, the Bernoulli random matrix ensemble, by E*. One has for these lim_{r to infinity}( lim_{n to infinity} (1/n) ln(E(perm_m(A))) -lim_{n to infinity} (1/n) ln(E*(perm_m(A))) ) = 0 and lim_{n to infinity} (1/n) ln(E(perm_m(A)perm_m'(A))) = lim_{n to infinity} (1/n) ln(E(perm_m(A))) + lim_{n to infinity} (1/n) ln(E(perm_m'(A))) Here and in all such formulas the subscripts m,m' are assumed proportional to n. It seems likely to us that lim_{r to infinity}( lim_{n to infinity} (1/n) ln(E*(perm_m(A)perm_m'(A))) - lim_{n to infinity} (1/n) ln(E*(perm_m(A))) + - lim_{n to infinity} (1/n) ln(E*(perm_m'(A))) ) = 0 We believe: "E gives us the `correct' expectations in these equations, and E* is only `correct' in the r to infinity limit."