17-54 Paul Federbush

On the Approximate Asymptotic Statistical Independence of the Permanents of 0-1 Matrices, I (21K, latex) May 3, 17

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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.

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