Domino tilings with diagonal impurities

TL;DR

This paper analyzes the dimer model on a specific lattice, providing an exact formula for impurity probabilities in domino tilings with impurities, linking combinatorics and random walk theory.

Contribution

It introduces an exact probabilistic formula for impurities in domino tilings on the square-octagon lattice, connecting dimer models with random walk analysis.

Findings

Derived an explicit formula for impurity probabilities

Connected domino tilings with random walk behavior

Enhanced understanding of impurity distribution in dimer models

Abstract

This paper studies the dimer model on the dual graph of the square-octagon lattice, which can be viewed as the domino tilings with impurities in some sense. In particular, under a certain boundary condition, we give an exact formula representing the probability of finding an impurity at a given site in a uniformly random dimer configuration in terms of simple random walks on the square lattice.