Stochastic Flips on Dimer Tilings

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.

💡 Research Summary

The paper investigates a stochastic process on dimer tilings of the triangular lattice, motivated by quasicrystal growth. A dimer is a lozenge made of two adjacent triangles, and a tiling of a finite simply‑connected domain with n dimers is considered. The authors define the “energy” E(ω) of a tiling ω as the total number of edges shared by two identical tiles (errors). An error‑free tiling has E=0 and can be visualized as a flat stepped surface in three dimensions. Errors form the boundaries of regions called islands (higher than the surrounding) or holes (lower). The volume V(ω) is defined as the sum of island areas minus the sum of hole areas; it equals the minimal number of flips required to reach an error‑free configuration.

A flip is a local rearrangement at a vertex where exactly three tiles meet; geometrically it adds or removes a unit cube in the three‑dimensional lift. Each flip changes the volume by ±1 and the energy by an element of {0, ±2, ±4, ±6}. The cooling process studied selects, uniformly at random, among all flips that do not increase the energy (ΔE ≤ 0). If no such flip exists, the process stops; the resulting “frozen” tiling must be error‑free (Proposition 1).

The main quantity of interest is the worst‑case expected convergence time
(b_T(n)=\max_{\omega\in\Omega_n}\mathbb{E}