An n-Dimensional Generalization of the Rhombus Tiling

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.

Physica A: Statistical Mechanics and its Applications, 2012

Reduced dimensionality in two dimensions is a topic of current interest. We use model systems to investigate the statistical mechanics of ideal networks. The tilings have possible applications such as the 2D locations of pore sites in nanoporous arrays (quantum dots), in the 2D hexagonal structure of graphene, and as adsorbates on quasicrystalline crystal surfaces. We calculate the statistical mechanics of these networks, such as the partition function, free energy, entropy, and enthalpy. The plots of these functions versus the number of links in the finite networks result in power law regression. We also determine the degree distribution, which is a combination of power law and rational function behavior. In the large-scale limit, the degree of these 2D networks approaches 3, 4, and 6, in agreement with the degree of the regular tilings. In comparison, a Penrose tiling has a degree also equal to about 4.

Journal of Statistical Physics, 2005

We perform numerical studies including Monte Carlo simulations of high rotational symmetry random tilings. For computational convenience, our tilings obey fixed boundary conditions in regular polygons. Such tilings are put in correspondence with algorithms for sorting lists in computer science. We obtain statistics on path counting and vertex coordination which compare well with predictions of mean-field theory and allow estimation of the configurational entropy, which tends to the value 0.568 per vertex in the limit of continuous symmetry. Tilings with phason strain appear to share the same entropy as unstrained tilings, as predicted by mean-field theory. We consider the thermodynamic limit and argue that the limiting fixed boundary entropy equals the limiting free boundary entropy, although these differ for finite rotational symmetry.

Journal of Physics A-mathematical and General, 1998

We introduce a concept for random tilings which, comprising the conventional one, is also applicable to tiling ensembles without height representation. In particular, we focus on the random tiling entropy as a function of the tile densities. In this context, and under rather mild assumptions, we prove a generalization of the first random tiling hypothesis which connects the maximum of

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.

Journal of Statistical Physics, 2002

Three-dimensional integer partitions provide a convenient representation of codimension-one three-dimensional random rhombus tilings. Calculating the entropy for such a model is a notoriously difficult problem. We apply transition matrix Monte Carlo simulations to evaluate their entropy with high precision. We consider both free-and fixed-boundary tilings. Our results suggest that the ratio of free-and fixed-boundary entropies is σ f ree /σ f ixed = 3/2, and can be interpreted as the ratio of the volumes of two simple, nested, polyhedra. This finding supports a conjecture by Linde, Moore and Nordahl concerning the "arctic octahedron phenomenon" in three-dimensional random tilings.

Physical Review E, 2008

We consider the construction of point processes from tilings, with equal volume tiles, of d-dimensional Euclidean space R d. We show that one can generate, with simple algorithms ascribing one or more points to each tile, point processes which are "superhomogeneous" (or "hyperuniform"), i.e., for which the structure factor S(k) vanishes when the wavenumber k tends to zero. The exponent of the leading small-k behavior, S(k → 0) ∝ k γ , depends in a simple manner on the nature of the correlation properties of the specific tiling and on the conservation of the mass moments of the tiles. Assigning one point to the center of mass of each tile gives the exponent γ = 4 for any tiling in which the shapes and orientations of the tiles are short-range correlated. Smaller exponents, in the range 4 − d < γ < 4 (and thus always superhomogeneous for d ≤ 4), may be obtained in the case that the latter quantities have long-range correlations. Assigning more than one point to each tile in an appropriate way, we show that one can obtain arbitrarily higher exponents in both cases. We illustrate our results with explicit constructions using known deterministic tilings, as well as some simple stochastic tilings for which we can calculate S(k) exactly. Our results provide, we believe, the first explicit analytical construction of point processes with γ > 4. Applications to condensed matter physics, and also to cosmology, are briefly discussed.

Duke Mathematical Journal, 1996

We prove an asymptotic formula for the probability that, if one chooses a domino tiling of a large Aztec diamond at random according to the uniform distribution on such tilings, the tiling will contain a domino covering a given pair of adjacent lattice squares. This formula quantifies the effect of the diamond's boundary conditions on the behavior of typical tilings; in addition, it yields a new proof of the arctic circle theorem of Jockusch, Propp, and Shor. Our approach is to use the saddle point method to estimate certain weighted sums of squares of Krawtchouk polynomials (whose relevance to domino tilings is demonstrated elsewhere), and to combine these estimates with some exponential sum bounds to deduce our final result. This approach generalizes straightforwardly to the case in which the probability distribution on the set of tilings incorporates bias favoring horizontal over vertical tiles or vice versa. We also prove a fairly general large deviation estimate for domino tilings of simply-connected planar regions that implies that some of our results on Aztec diamonds apply to many other similar regions as well.

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

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.