Stochastic Flips on Dimer Tilings

TL;DR

This paper studies a Markov process on dimer tilings inspired by quasicrystal growth, providing an upper bound on the expected convergence time to fixed points and discussing the gap with previous bounds.

Contribution

It introduces a new Markov process model for quasicrystal growth on dimer tilings and proves an improved upper bound on the convergence time.

Findings

Proved an O(n^2.5) upper bound on expected flips to reach fixed points.

Numerical experiments suggest the bound is close to quadratic in n.

Discussed the gap between the new bound and previous results.

Abstract

This paper introduces a Markov process inspired by the problem of quasicrystal growth. It acts over dimer tilings of the triangular grid by randomly performing local transformations, called {\em flips}, which do not increase the number of identical adjacent tiles (this number can be thought as the tiling energy). Fixed-points of such a process play the role of quasicrystals. We are here interested in the worst-case expected number of flips to converge towards a fixed-point. Numerical experiments suggest a bound quadratic in the number n of tiles of the tiling. We prove a O(n^2.5) upper bound and discuss the gap between this bound and the previous one. We also briefly discuss the average-case.