Tiled partial cubes

In the quest to better understand the connection between median graphs, triangle‐free graphs and partial cubes, a hierarchy of subclasses of partial cubes has been introduced. In this article, we study the role of tiled partial cubes in this scheme. For instance, we prove that almost‐median graphs are tiled and that tiled partial cubes are semi‐median. We also describe median graphs as tiled partial cubes without convex Q and extend an inequality for median graphs to a larger subclass of partial cubes. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 91–103, 2002     

THE SCENERY FLOWFOR GEOMETRIC STRUCTURES ON THE TORUS: THE LINEAR SETTING

The authors define the scenery flow of the torus. The flow space is the union of all flat 2-dimensional tori of area $1$ with a marked direction (or equivalently, the union of all tori with a quadratic differential of norm 1). This is a $5$-dimensional space, and the flow acts by following individual points under an extremal deformation of the quadratic differential. The authors define associated horocycle and translation flows; the latter preserve each torus and are the horizontal and vertical flows of the corresponding quadratic differential. The scenery flow projects to the geodesic flow on the modular surface, and admits, for each orientation preserving hyperbolic toral automorphism, an invariant $3$-dimensional subset on which it is the suspension flow of that map. The authors first give a simple algebraic definition in terms of the group of affine maps of the plane, and prove that the flow is Anosov. They give an explicit formula for the first-return map of the flow on convenient cross-sections. Then, in the main part of the paper, the authors give several different models for the flow and its cross-sections, in terms of: \item{$\bullet$} stacking and rescaling periodic tilings of the plane; \item{$\bullet$} symbolic dynamics: the natural extension of the recoding of Sturmian sequences, or the $S$-adic system generated by two substitutions; \item{$\bullet$} zooming and subdividing quasi-periodic tilings of the real line, or aperiodic quasicrystals of minimal complexity; \item{$\bullet$} the natural extension of two-dimensional continued fractions; \item{$\bullet$} induction on exchanges of three intervals; \item{$\bullet$} rescaling on pairs of transverse measure foliations on the torus, or the Teichm\"uller flow on the twice-punctured torus.     

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.     

Dimers, tilings and trees

Generalizing results of Temperley (London Mathematical Society Lecture Notes Series 13 (1974) 202), Brooks et al. (Duke Math. J. 7 (1940) 312) and others (Electron. J. Combin. 7 (2000); Israel J. Math. 105 (1998) 61) we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees of the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to “discrete analytic functions” on the bipartite graph.The equivalence is extended to infinite periodic graphs, and we classify the resulting “almost periodic” tilings and harmonic functions.     

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.     

Partitions of the phase space of a measure preserving ℤ d -action into towers

Alpern proved that the phase space of an aperiodic measure preserving automorphismT can be decomposed into Rokhlin-Halmos towers of given heightsh _i and weightsm _i whenever the numbersh _i are relatively prime. In this paper an extension of the Alpern theorem to the case of free ℤ ^d -actions is given. Namely, we investigate the decomposability of the action phase space into towers of rectangular, form and present conditions on the configuration (the set of tower forms) sufficient for the existence of such a decomposition. The proof of the main result uses the technique of tilings. Translated fromMatematicheskie Zametki, Vol. 65, No. 5, pp. 712–725, May, 1999.     

A microscopic model with quasicrystalline properties

A classical lattice gas model with two-body nearest neighbor interactions and without periodic ground-state configurations is presented. The main result is the existence of a decreasing sequence of temperatures for which the Gibbs states have arbitrarily long periods. It is possible that the sequence accumulates at nonzero temperature, giving rise to a quasiperiodic equilibrium state.     

Enumeration of Lozenge Tilings of Hexagons with Cut-Off Corners

Motivated by the enumeration of a class of plane partitions studied by Proctor and by considerations about symmetry classes of plane partitions, we consider the problem of enumerating lozenge tilings of a hexagon with “maximal staircases” removed from some of its vertices. The case of one vertex corresponds to Proctor's problem. For two vertices there are several cases to consider, and most of them lead to nice enumeration formulas. For three or more vertices there do not seem to exist nice product formulas in general, but in one special situation a lot of factorization occurs, and we pose the problem of finding a formula for the number of tilings in this case.     

The Problem of the Pawns

In this paper we study the number M _m;n of ways to place nonattacking pawns on an m x n chessboard. We find an upper bound for M _m;n and consider a lower bound for M _m;n by reducing this problem to that of tiling an (m+1)x(n+1) board with square tiles of size 1x1 and 2x2. Also, we use the transfer-matrix method to implement an algorithm that allows us to get an explicit formula for M _m;n for given m. Moreover, we show that the double limit := lim _m;n (M _m;n ) ^1/mn exists and 2.25915263 n 2.26055675. Also, the true value of n is around 2.2591535382327...AMS Subject Classification: 05A16, 05C50, 52C20, 82B20.     

A remark on Schrödinger operators on aperiodic tilings

We prove that for a large class of Schrödinger operators on aperiodic tilings the spectrum and the integrated density of states are the same for all tilings in the local isomorphism class, i.e., for all tilings in the orbit closure of one of the tilings. This generalizes the argument in earlier work from discrete strictly ergodic operators onl ²( ^d ) to operators on thel ²-spaces of sets of vertices of strictly ergodic tilings.