A bijection theorem for domino tiling with diagonal impurities

TL;DR

This paper establishes a bijection between dimer coverings with impurities on a non-bipartite graph and spanning forests, extending the Temperley bijection, and analyzes impurity distribution and local move connectivity.

Contribution

It introduces a novel bijection extending the Temperley bijection to non-bipartite graphs with impurities and proves local move connectedness.

Findings

Impurities tend to distribute on the boundary as indicated by simulations.

A bijection between dimer coverings and spanning forests is established.

Bounds on the number of dimer coverings and impurity probabilities are derived.

Abstract

We consider the dimer problem on a non-bipartite graph $G$, where there are two types of dimers one of which we regard impurities. Results of simulations using Markov chain seem to indicate that impurities are tend to distribute on the boundary, which we set as a conjecture. We first show that there is a bijection between the set of dimer coverings on $G$ and the set of spanning forests on two graphs which are made from $G$, with configuration of impurities satisfying a pairing condition. This bijection can be regarded as a extension of the Temperley bijection. We consider local move consisting of two operations, and by using the bijection mentioned above, we prove local move connectedness. We further obtained some bound of the number of dimer coverings and the probability finding an impurity at given edge, by extending the argument in our previous result.