Related papers: Flip dynamics in octagonal rhombus tiling sets
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…
We prove that the continuous-time, single-flip Glauber dynamics for lozenge tilings of the size-$N$ hexagon mix in time $N^{2+o(1)}$. This was predicted to hold on fairly general domains of diameter $N$ (on the basis of the ``Lifshitz law''…
Some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied. A geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems, thus…
The assembly of molecular networks into structures such as random tilings and glasses has recently been demonstrated for a number of two-dimensional systems. These structures are dynamically-arrested on experimental timescales so the…
A \textit{domino} is a $2\times 1\times 1$ parallelepiped formed by the union of two unit cubes and a \textit{slab} is a $2\times 2\times 1$ parallelepiped formed by the union of four unit cubes. We are interested in tiling regions formed…
We investigate the dynamics of tiling dynamical systems and their deformations. If two tiling systems have identical combinatorics, then the tiling spaces are homeomorphic, but their dynamical properties may differ. There is a natural map…
Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural…
The rhombus tilings of a simply connected domain of the Euclidean plane are known to form a flip-connected space (a flip is the elementary operation on rhombus tilings which rotates 180{\deg} a hexagon made of three rhombi). Motivated by…
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…
Three-dimensional icosahedral random tilings are studied in the semi-entropic model. We introduce a global energy measure defined by the variance of the quasilattice points in orthogonal space. The specific heat shows a pronounced Schottky…
TThe prototypical problem we study here is the following. Given a $2L\times 2L$ square, there are approximately $\exp(4KL^2/\pi )$ ways to tile it with dominos, i.e. with horizontal or vertical $2\times 1$ rectangles, where $K\approx 0.916$…
We investigate tilings of cubiculated regions with two simply connected floors by 2 x 1 x 1 bricks. More precisely, we study the flip connected component for such tilings, and provide an algebraic invariant that "almost" characterizes the…
The broad motivation of this work is a rigorous understanding of reversible, local Markov dynamics of interfaces, and in particular their speed of convergence to equilibrium, measured via the mixing time $T_{mix}$. In the…
Given a finite set ${S_1...,S_k}$ of substitution maps acting on a certain finite number (up to translations) of tiles in $\rd$, we consider the multi-substitution tiling space associated to each sequence $\bar a\in {1,...,k}^{\mathbb{N}}$.…
We study a reversible continuous-time Markov dynamics on lozenge tilings of the plane, introduced by Luby et al. Single updates consist in concatenations of $n$ elementary lozenge rotations at adjacent vertices. The dynamics can also be…
We study transition matrices for projected dynamics in the energy-magnetization space, magnetization space and energy space. Several single spin flip dynamics are considered such as the Glauber and Metropolis canonical ensemble dynamics and…
A particular, two-dimensional, tiling model, composed by the so called Wang tiles has been studied at finite temperature by Monte Carlo numerical simulations. In absence of any thermal bath the Wang tiles give the opportunity of building a…
We construct random dynamics on collections of non-intersecting planar contours, leaving invariant the distributions of length- and area-interacting polygonal Markov fields with V-shaped nodes. The first of these dynamics is based on the…
Two-dimensional random tilings of rhombi can be seen as projections of two-dimensional membranes embedded in hypercubic lattices of higher dimensional spaces. Here, we consider tilings projected from a $D$-dimensional space. We study the…
We have recently shown [Blunt et al., Science 322, 1077 (2008)] that p-terphenyl-3,5,3',5'-tetracarboxylic acid adsorbed on graphite self-assembles into a two-dimensional rhombus random tiling. This tiling is close to ideal, displaying long…