A BK inequality for random matchings

TL;DR

This paper proves a BK inequality for the symmetric difference of vertices covered by a uniform random matching in bipartite graphs, extending the understanding of correlation inequalities in combinatorics.

Contribution

It establishes a BK inequality for the symmetric difference of vertex sets in uniform random matchings on bipartite graphs, a novel extension in combinatorial probability.

Findings

Proves BK inequality for symmetric difference in bipartite matchings

Extends correlation inequalities to new combinatorial structures

Provides a theoretical foundation for analyzing random matchings

Abstract

Let $G=(S,T,E)$ be a bipartite graph. For a matching $M$ of $G$, let $V(M)$ be the set of vertices covered by $M$, and let $B(M)$ be the symmetric difference of $V(M)$ and $S$. We prove that if $M$ is a uniform random matching of $G$, then $B(M)$ satisfies the BK inequality for increasing events.