Stochastic Flips on Dimer Tilings

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.