Vacancy localization in the square dimer model

TL;DR

This paper investigates vacancy movement in the square dimer model, revealing probabilistic localization, domain size distributions, and diffusion properties through graph-theoretic methods and simulations.

Contribution

It introduces a graph-theoretic classification for dimer configurations with a vacancy, extending spanning tree methods to analyze vacancy mobility and localization.

Findings

57/4 - 10√2 probability of vacancy being jammed

Power law decay with exponent 9/8 for accessible domain size

Vacancy reachability probability decreases as system size^{-1/4}

Abstract

We study the classical dimer model on a square lattice with a single vacancy by developing a graph-theoretic classification of the set of all configurations which extends the spanning tree formulation of close-packed dimers. With this formalism, we can address the question of the possible motion of the vacancy induced by dimer slidings. We find a probability 57/4-10Sqrt[2] for the vacancy to be strictly jammed in an infinite system. More generally, the size distribution of the domain accessible to the vacancy is characterized by a power law decay with exponent 9/8. On a finite system, the probability that a vacancy in the bulk can reach the boundary falls off as a power law of the system size with exponent 1/4. The resultant weak localization of vacancies still allows for unbounded diffusion, characterized by a diffusion exponent that we relate to that of diffusion on spanning trees. We also implement numerical simulations of the model with both free and periodic boundary conditions.