# On the Approximate Asymptotic Statistical Independence of the Permanents of 0-1 Matrices

## TL;DR

This paper investigates the approximate statistical independence of permanents of 0-1 matrices with fixed row and column sums, providing new bounds and computational evidence for the approximation measure E_1.

## Contribution

It proves a tighter asymptotic independence result for the measure E_1 compared to the measure E, with improved error bounds and computational validation.

## Key findings

- E_1 measure closely approximates E with an error of O(1/n^2)
- Computational evidence supports the asymptotic independence under E_1
- New theoretical bounds improve understanding of permanents in structured matrices

## 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 know from the work of Wanless and Pernici that E(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E(perm_{m_i}(A)) * (1+ O(1/n^4)) 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)) 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. We have computer evidence for E_1(prod_{i=1}^N (perm_{m_i}(A)) = prod_{i=1}^N (E_1(perm_{m_i}(A)) * (1+ O(1/n^4)).