Law of Large Numbers for Permanents of Random Constrained Matrices

TL;DR

This paper establishes a law of large numbers for the permanents of constrained random matrices with a fixed number of zeros per row, showing concentration around the van der Waerden bound under certain growth conditions.

Contribution

It introduces new conditions for the weak law of large numbers for permanents of constrained matrices with random zero patterns and positive entries.

Findings

Permanent concentrates around the van der Waerden bound when zeros grow faster than √n.

Sufficient conditions for weak convergence of scaled permanents are provided.

Results extend understanding of permanents for constrained random matrices.

Abstract

Permanents of random matrices with independent and identically distributed (i.i.d.) entries have extensively studied in literature and convergence and concentration properties are known under varying assumptions on the distributions. In this paper we consider constrained~\(n \times n\) random matrices where each row has a deterministic number~\(r = r(n)\) zero entries and the rest of the entries are independent random variables that are positive almost surely. The positions of the zeros within each row are also random and we establish sufficient conditions for the existence of a weak law of large numbers for the permanent, appropriately scaled and centred. As a special case, we see that if~\(r\) grows faster than~\(\sqrt{n},\) then the permanent of a randomly chosen constrained~\(0-1\) matrix is concentrated around the van der Waerden bound with high probability, i.e. with probability converging to one as~\(n \rightarrow \infty.\)