Hitting Matrix and Domino Tiling with Diagonal Impurities

TL;DR

This paper extends domino tiling analysis to include boundary impurities, providing methods to count perfect matchings and studying impurity placement probabilities in large graphs.

Contribution

It introduces the hitting matrix method for impurity counting and offers an alternative proof using Kenyon-Wilson's grove formula, advancing understanding of impurity effects in tilings.

Findings

Counting perfect matchings with multiple boundary impurities using hitting matrix

Alternative proof of previous results via grove counting formula

Analysis of impurity placement probabilities and scaling limits

Abstract

As a continuation to our previous work [9, 10], we consider the domino tiling problem with impurities. (1) if we have more than two impurities on the boundary, we can compute the number of corresponding perfect matchings by using the hitting matrix method[4]. (2) we have an alternative proof of the main result in [9] and result in (1) above using the formula by Kenyon-Wilson [6, 7] of counting the number of groves on the circular planar graph. (3) we study the behavior of the probability of finding the impurity at a given site when the size of the graph tends to infinity, as well as the scaling limit of those.