Spaces of domino tilings

We consider the set of all tilings by dominoes (2×1 rectangles) of a surface, possibly with boundary, consisting of unit squares. Convert this set into a graph by joining two tilings by an edge if they differ by aflip, i.e., a 90° rotation of a pair of side-by-side dominoes. We give a criterion to decide if two tilings are in the same connected component, a simple formula for distances, and a method to construct geodesics in this graph. For simply connected surfaces, the graph is connected. By naturally adjoining to this graph higher-dimensional cells, we obtain a CW-complex whose connected components are homotopically equivalent to points or circles. As a consequence, for any region different from a torus or Klein bottle, all geodesics with common endpoints are equivalent in the following sense. Build a graph whose vertices are these geodesics, adjacent if they differ only by the order of two flips on disjoint squares: this graph is connected. The first two authors received support from SCT and CNPq, Brazil. The other two were supported by a grant for undergraduates by CNPq.     

A variational principle for domino tilings

We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within (for an appropriate metric) of the unique entropy-maximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby resolving an issue first raised by Kasteleyn. Our methods also apply to dimer models on other grids and their associated tiling models, such as tilings of the plane by three orientations of unit lozenges.

     

Coupling functions for domino tilings of Aztec diamonds

The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics. In this paper, we consider three different weightings of domino tilings of the Aztec diamond and show using recurrence relations, that we can compute the inverse Kasteleyn matrix. These weights are the one-periodic weighting where the horizontal edges have one weight and the vertical edges have another weight, the q^vol$q^{\mathrm{vol}}$ weighting which corresponds to multiplying the product of tile weights by q if we add a ‘box’ to the height function and the two-periodic weighting which exhibits a flat region with defects in the center.     

Enumeration of domino tilings of an Aztec rectangle with boundary defects

In this paper we enumerate domino tilings of an Aztec rectangle with arbitrary defects of size one on all boundary sides. This result extends previous work by different authors: Mills–Robbins–Rumsey and Elkies–Kuperberg–Larsen–Propp. We use the method of graphical condensation developed by Kuo and generalized by Ciucu, to prove our results; a common generalization of both Kuo's and Ciucu's result is also presented here.     

A note on the structure of spaces of domino tilings

We study spaces of tilings, formed by tilings which are on a geodesic between two fixed tilings of the same domain (the distance is defined using local flips). We prove that each space of tilings is homeomorphic to an interval of tilings of a domain when flips are classically directed by height functions.     

Local connectedness in transformation groups

This note shows that under very general conditions, in a topological transformation group, the natural map from the group onto an orbit is almost open. The implications for local connectedness are investigated. In particular, if the image of a path component of the group is sufficiently ``robust", the orbit will be locally connected.

     

Local connectedness in merotopic spaces

Abstract

In this paper uniformly locally uniformly connected merotopic spaces are studied. It turns out that their structural behaviour is essentially similar to that one of locally connected topological spaces. The introduced concept is also investigated for spaces of functions between filter-merotopic spaces (e.g. topological spaces, proximity spaces, convergence spaces) and the relationship to other concepts of local connectedness is clarified. In particular, the category of uniformly locally uniformly connected filter-merotopic spaces is Cartesian closed.     

Laurent biorthogonal polynomials, q-Narayana polynomials and domino tilings of the Aztec diamonds

A Toeplitz determinant whose entries are described by a q-analogue of the Narayana polynomials is evaluated by means of Laurent biorthogonal polynomials which allow of a combinatorial interpretation in terms of Schröder paths. As an application, a new proof is given to the Aztec diamond theorem by Elkies, Kuperberg, Larsen and Propp concerning domino tilings of the Aztec diamonds. The proof is based on the correspondence with non-intersecting Schröder paths developed by Johansson.     

Local rules and global order, or aperiodic tilings

Can local rules impose a global order? If yes, when and how? This is a philosophical question that could be asked in many cases. How does local interaction of atoms create crystals (or quasicrystals)? How does one living cell manage to develop into a pine cone whose seeds form spirals (and the number of spirals     

Local connectedness and extension of uniformly continuous functions

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. The straight spaces have been studied in [A. Berarducci, D. Dikranjan, J. Pelant, An additivity theorem for uniformly continuous functions, Topology and its Applications 146-147 (2005) 339-352], which contains characterization of the straight spaces within the class of the locally connected spaces (they are the uniformly locally connected ones) and the class of the totally disconnected spaces (they coincide with the totally disconnected Atsuji spaces). We show that the completion of a straight space is straight and we characterize the dense straight subspaces of a straight space. In order to clarify further the relation between straightness and the level of local connectedness of the space we introduce two more intermediate properties between straightness and uniform local connectedness and we give various examples to distinguish them. One of these properties coincides with straightness for complete spaces and provides in this way a useful characterization of complete straight spaces in terms of the behaviour of the quasi-components of the space.     

Square tilings with prescribed combinatorics

LetT be a triangulation of a quadrilateralQ, and letV be the set of vertices ofT. Then there is an essentially unique tilingZ=(Z_v: v ∈ V) of a rectangleR by squares such that for every edge <u,v> ofT the corresponding two squaresZ _u, Z_vare in contact and such that the vertices corresponding to squares at corners ofR are at the corners ofQ. It is also shown that the sizes of the squares are obtained as a solution of an extremal problem which is a discrete version of the concept of extremal length from conformal function theory. In this discrete version of extremal length, the metrics assign lengths to the vertices, not the edges. A practical algorithm for computing these tilings is presented and analyzed. The author thankfully acknowledges support of NSF grant DMS-9112150.     

Local tuning and partition strategies for diagonal GO methods

Summary.  In this paper, global optimization (GO) Lipschitz problems are considered where the multi-dimensional multiextremal objective function is determined over a hyperinterval. An efficient one-dimensional GO method using local tuning on the behavior of the objective function is generalized to the multi-dimensional case by the diagonal approach using two partition strategies. Global convergence conditions are established for the obtained diagonal geometric methods. Results of a wide numerical comparison show a strong acceleration reached by the new methods working with estimates of the local Lipschitz constants over different subregions of the search domain in comparison with the traditional approach. Received July 13, 2001 / Revised version received March 14, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 65K05, 90C30 Correspondence to: Yaroslav D. Sergeyev     

Substitution tilings with statistical circular symmetry

Two new series of substitution tilings are introduced in which the tiles appear in infinitely many orientations. It is shown that several properties of the well-known pinwheel tiling do also hold for these new examples, and, in fact, for all primitive substitution tilings showing tiles in infinitely many orientations.     

Domino tilings of rectangles with fixed width

Let t(k,n) denote the number of ways to tile a 1 × n rectangle with 1 × 2 rectangles (called dominoes). We show that for each fixed k the sequence t_k=(t(k,0), t(k,1),…) satisfies a difference equation (linear, homogeneous, and with constant coefficients). Furthermore, a computational method is given for finding this difference equation together with the initial terms of the sequence. This gives rise to a new way to compute t(k,n) which differs completely with the known Pfaffian method. The generating function of t_k is a rational function F_k, and F_k is given explicitly for k=1,…,8. We end with some conjectures concerning the form of F_k based on our computations.     

Lozenge tilings of hexagons with arbitrary dents

Eisenkölbl gave a formula for the number of lozenge tilings of a hexagon on the triangular lattice with three unit triangles removed from along alternating sides. In earlier work, the first author extended this to the situation when an arbitrary set of unit triangles is removed from along alternating sides of the hexagon. In this paper we address the general case when an arbitrary set of unit triangles is removed from along the boundary of the hexagon.     

n-Homeohedral types of tilings

For each n, there are only finitely many topological types of normal n-homeohedral tilings, and for every such type there is an n-isohedral representative which displays essentially the same symmetries.     

Affability of Euclidean tilings

We prove that every minimal equivalence relation on a Cantor set arising from the continuous hull of an aperiodic and repetitive Euclidean tiling is affable. To cite this article: F. Alcalde Cuesta et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).