Properties of random tilings in three dimensions

Physical Review E, 2010

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 dynamicallyarrested on experimental timescales so the critical regime in their formation is that of initial growth. Here we identify a transition from energetic to entropic stabilisation in the nucleation and growth of a molecular rhombus tiling. Calculations based on a lattice gas model show that clustering of topological defects and the formation of faceted boundaries followed by a slow relaxation to equilibrium occurs under conditions of energetic stabilisation. We also identify an entropicallystabilised regime in which the system grows directly into an equilibrium configuration without the need for further relaxation. Our results provide a methodology for identifying equilibrium and nonequilibrium randomness in the growth of molecular tilings, and we demonstrate that equilibrium spatial statistics are compatible with exponentially slow dynamical behaviour.

1970

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 limiting case, when the quantity D, and therefore the number of different species of tiles, become large. We had previously demonstrated [1] that, in this limit, the thermodynamic properties of the tiling become independent of the boundary conditions. The exact value of the limiting entropy and finite D corrections remain open questions. Here, we develop a meanfield theory, which uses an iterative description of the tilings based on an analogy with avoiding oriented walks on a random tiling. We compare the quantities so-obtained with numerical calculations. We also discuss the role of spatial correlations.

Physical Review E, 2013

We consider the domino tilings of an Aztec diamond with a cutoff corner of macroscopic square shape and given size, and address the bulk properties of tilings as the size is varied. We observe that the free energy exhibits a third-order phase transition when the cutoff square, increasing in size, reaches the arctic ellipse-the phase separation curve of the original (unmodified) Aztec diamond. We obtain this result by studying the thermodynamic limit of certain nonlocal correlation function of the underlying six-vertex model with domain wall boundary conditions, the so-called emptiness formation probability (EFP). We consider EFP in two different representations: as a tau-function for Toda chains and as a random matrix model integral. The latter has a discrete measure and a linear potential with hard walls; the observed phase transition shares properties with both Gross-Witten-Wadia and Douglas-Kazakov phase transitions.

Journal of Statistical Physics, 2005

We study random tiling models in the limit of high rotational symmetry. In this limit a mean-field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions. We base our mean-field theory on an iterative tiling construction inspired by the work of de Bruijn. In addition to the entropy, we consider correlation functions, phason elasticity and the thermodynamic limit. Tilings of dimension other than two are considered briefly.

Physical Review Letters, 1990

A three-dimensional random-tiling icosahedral quasicrystal is studied by a Monte Carlo simulation. The hypothesis of long-range positional order in the system is confirmed through analysis of the finitesize scaling behavior of phason fluctuations and Fourier peak intensities. By investigating the diffuse scattering we determine the phason stiffness constants. A finite-size scaling form for the Fourier intensity near an icosahedral reciprocal wave vector is proposed.

Journal of Non-Crystalline Solids, 2004

We calculate the configurational entropy of codimension-one three-dimensional random rhombus tilings. We use threedimensional integer partitions to represent these tilings. We apply transition matrix Monte Carlo simulations to evaluate their entropy with high precision. We explore free-as well as fixed-boundary conditions and our numerical results suggest that the ratio of free-and fixed-boundary entropies is r free =r fixed ¼ 3=2, and can be interpreted as the ratio of the volumes of two simple, nested, polyhedra. This ratio confirms a conjecture by Linde et al. concerning the Ôarctic octahedron phenomenon' in three-dimensional codimension-one random tilings.

2001

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 generalizing previous results on the more restricted class of codimension-one tilings. In particular this method gives access to counting formulas, which are directly related to questions of entropy in these statistical systems. Methods and tools from the field of enumerative combinatorics are used.

Physical Review Letters, 1996

We consider the two-dimensional random tiling model introduced by Cockayne, i.e., the ensemble of all possible coverings of the plane without gaps or overlaps with squares and various hexagons. At the appropriate relative densities the correlations have eightfold rotational symmetry. We reformulate the model in terms of a random tiling ensemble with identical rectangles and isosceles triangles. The partition function of this model can be calculated by diagonalizing a transfer matrix using the Bethe Ansatz (BA). The BA equations can be solved providing exact values of the entropy and elastic constants.

The European Physical Journal E, 2010

A vast range of both living and inanimate planar cellular partitions obeys universal empirical laws describing their structure. To better understand this observation, we analyze the morphometric parameters of a sizeable set of experimental data that includes animal and plant tissues, patterns in desiccated starch slurry, suprafroth in type-I superconductors, soap froths, and geological formations. We characterize the tilings by the distributions of polygon reduced area, a scale-free measure of the roundedness of polygons. These distributions are fairly sharp and seem to belong to the same family. We show that the experimental tilings can be mapped onto the model tilings of equal-area, equal-perimeter polygons obtained by numerical simulations. This suggests that the random two-dimensional patterns can be parametrized by their median reduced area alone.

2021

Jerome Lloyd, 2, 3, ∗ Sounak Biswas, ∗ Steven H. Simon, S. A. Parameswaran, and Felix Flicker 4 Rudolf Peierls Centre for Theoretical Physics, Parks Road, Oxford OX1 3PU, United Kingdom School of Physics and Astronomy, University of Birmingham, Edgbaston Park Road, Birmingham, B15 2TT, United Kingdom Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße, Dresden, 01187, Germany School of Physics and Astronomy, Cardiff University, The Parade, Cardiff CF24 3AA, United Kingdom