Local Statistics of the $M_n$-Dimer Model

TL;DR

This paper investigates the local statistics and correlations of the $M_n$-dimer model, a probabilistic extension of the classical dimer model, on bipartite graphs with matrix weights, providing formulas and classification of local moves.

Contribution

It introduces formulas for local edge statistics in the $M_n$-dimer model and classifies local moves to simplify analysis on bipartite graphs.

Findings

Derived formulas for local edge statistics and correlations.

Classified local moves for simplifying graph analysis.

Extended understanding of the $M_n$-dimer model's probabilistic structure.

Abstract

The classical dimer model is concerned with the (weighted) enumeration of perfect matchings of a graph. An $n$-dimer cover is a multiset of edges that can be realized as the disjoint union of $n$ individual matchings. For a probability measure recently defined by Douglas, Kenyon, and Shi, which we call the $M_n$-dimer model, we study random $n$-dimer covers on bipartite graphs with matrix edge weights and produce formulas for local edge statistics and correlations. We also classify local moves that can be used to simplify the analysis of such graphs.