Mixing Time on the Kagome Lattice

TL;DR

This paper analyzes the mixing time of a Markov chain on Kagome lattice tilings, providing polynomial upper bounds and suggesting long-range phenomena through simulations.

Contribution

It establishes $ ext{O}(N^4)$ upper bounds on mixing times for weighted and non-weighted Kagome tilings, and proves flip-connectivity for two prototile cases.

Findings

Polynomial mixing time bounds of $ ext{O}(N^4)$.

Flip-connectivity of Kagome tilings with two prototiles.

Simulations indicating possible long-range correlations.

Abstract

We consider tilings of a closed region of the Kagome lattice (partition of the plane into regular hexagons and equilateral triangles such that each edge is shared by one triangle and one hexagon). We are interested in the rate of convergence to the stationarity of a natural Markov chain defined on the set of Kagome tilings. The rate of convergence can be represented by the mixing time which mesures the amount of time it takes the chain to be close to its stationary distribution. We obtain a $\mathcal{O}(N^4)$ upper bound on the mixing time of a weighted version of the natural Markov chain. We also consider Kagome tilings restrained to two prototiles, prove flip-connectivity and draw a $\mathcal{O}(N^4)$ upper bound as well on the mixing time of the natural Markov chain in a general (non weighted) case. Finally, we present simulations that suggest existence of a long range phenomenon.