Paul Federbush
On the Approximate Asymptotic Statistical Independence
of the Permanents of 0-1 Matrices, I
(21K, latex)
ABSTRACT. We consider the ensemble of n x n 0 - 1 matrices with all column and row sums
equal r. We give this ensemble the uniform weighting to construct a measure E.
We conjecture
E(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E(perm_{m_i}(A)) * (1+ O(1/n^3))
In this paper we prove
E_1(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E_1(perm_{m_i}(A)) * (1+ O(1/n^2))
and in the sequel paper, number II in this sequence, we show
E_1(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E_1(perm_{m_i}(A)) * (1+ O(1/n^3))
where E_1 is the measure constructed on the ensemble of n x n 0 - 1 matrices with
non-negative integer entries realized as the sum of r random permutation matrices.
E_1 is often used as an ``approximation'' to E, and the truth of the last equation
explains why we support the conjecture.