Flip dynamics in three-dimensional random tilings

TL;DR

This paper investigates the connectivity and dynamics of three-dimensional rhombus tilings under flip operations, proving connectivity in most cases and analyzing how cycles affect diffusion.

Contribution

It provides a rigorous proof of tiling set connectivity for codimension one and two, and explores the impact of cycles on flip dynamics through numerical simulations.

Findings

Sets of tilings of codimension one and two are connected for any dimension and size.

Cycles do not significantly slow down flip-assisted diffusion.

Connectivity depends on edge orientations in higher codimension tilings.

Abstract

We study single-flip dynamics in sets of three-dimensional rhombus tilings with fixed polyhedral boundaries. This dynamics is likely to be slowed down by so-called ``cycles'': such structures arise when tilings are encoded via the ``partition-on-tiling'' method and are susceptible to break connectivity by flips or at least ergodicity, because they locally suppress a significant amount of flip degrees of freedom. We first address the so-far open question of the connectivity of tiling sets by elementary flips. We prove exactly that sets of tilings of codimension one and two are connected for any dimension and tiling size. For higher-codimension tilings of dimension 3, the answer depends on the precise choice of the edge orientations, which is a non-trivial issue. In most cases, we can prove connectivity despite the existence of cycles. In the few remaining cases, among which the icosahedral symmetry, the question remains open. We also study numerically flip-assisted diffusion to explore the possible effects of the previously mentioned cycles. Cycles do not seem to slow down significantly the dynamics, at least as far as self-diffusion is concerned.