Flip dynamics in octagonal rhombus tiling sets

Abstract

We investigate the properties of classical single flip dynamics in sets of two-dimensional random rhombus tilings. Single flips are local moves involving 3 tiles which sample the tiling sets {\em via} Monte Carlo Markov chains. We determine the ergodic times of these dynamical systems (at infinite temperature): they grow with the system size $N_T$ like $Cst. N_T^2 \ln N_T$; these dynamics are rapidly mixing. We use an inherent symmetry of tiling sets and a powerful tool from probability theory, the coupling technique. We also point out the interesting occurrence of Gumbel distributions.