 1632 Paul Federbush
 Asymptotic Behavior of the Expectation Value of Permanent Products, a Sequel
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Apr 12, 16

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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 01 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.''
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