47 research outputs found

    Fast domino tileability

    Get PDF
    Domino tileability is a classical problem in Discrete Geometry, famously solved by Thurston for simply connected regions in nearly linear time in the area. In this paper, we improve upon Thurston's height function approach to a nearly linear time in the perimeter.Comment: Appeared in Discrete Comput. Geom. 56 (2016), 377-39

    An Overview of Domino and Lozenge Tilings

    Get PDF
    We consider tilings of quadriculated regions by dominoes and oftriangulated regions by lozenges. We present an overview of results concerning tileability, enumeration and the structure of the space of tilings

    Partition function of periodic isoradial dimer models

    Full text link
    Isoradial dimer models were introduced in \cite{Kenyon3} - they consist of dimer models whose underlying graph satisfies a simple geometric condition, and whose weight function is chosen accordingly. In this paper, we prove a conjecture of \cite{Kenyon3}, namely that for periodic isoradial dimer models, the growth rate of the toroidal partition function has a simple explicit formula involving the local geometry of the graph only. This is a surprising feature of periodic isoradial dimer models, which does not hold in the general periodic dimer case \cite{KOS}.Comment: 12 pages, 2 figure

    Components of domino tilings under flips in quadriculated cylinder and torus

    Full text link
    In a region RR consisting of unit squares, a domino is the union of two adjacent squares and a (domino) tiling is a collection of dominoes with disjoint interior whose union is the region. The flip graph T(R)\mathcal{T}(R) is defined on the set of all tilings of RR such that two tilings are adjacent if we change one to another by a flip (a 9090^{\circ} rotation of a pair of side-by-side dominoes). It is well-known that T(R)\mathcal{T}(R) is connected when RR is simply connected. By using graph theoretical approach, we show that the flip graph of 2m×(2n+1)2m\times(2n+1) quadriculated cylinder is still connected, but the flip graph of 2m×(2n+1)2m\times(2n+1) quadriculated torus is disconnected and consists of exactly two isomorphic components. For a tiling tt, we associate an integer f(t)f(t), forcing number, as the minimum number of dominoes in tt that is contained in no other tilings. As an application, we obtain that the forcing numbers of all tilings in 2m×(2n+1)2m\times (2n+1) quadriculated cylinder and torus form respectively an integer interval whose maximum value is (n+1)m(n+1)m

    Quadri-tilings of the plane

    Full text link
    We introduce {\em quadri-tilings} and show that they are in bijection with dimer models on a {\em family} of graphs {R}\{R^*\} arising from rhombus tilings. Using two height functions, we interpret a sub-family of all quadri-tilings, called {\em triangular quadri-tilings}, as an interface model in dimension 2+2. Assigning "critical" weights to edges of RR^*, we prove an explicit expression, only depending on the local geometry of the graph RR^*, for the minimal free energy per fundamental domain Gibbs measure; this solves a conjecture of \cite{Kenyon1}. We also show that when edges of RR^* are asymptotically far apart, the probability of their occurrence only depends on this set of edges. Finally, we give an expression for a Gibbs measure on the set of {\em all} triangular quadri-tilings whose marginals are the above Gibbs measures, and conjecture it to be that of minimal free energy per fundamental domain.Comment: Revised version, minor changes. 30 pages, 13 figure

    Statistical mechanics on isoradial graphs

    Get PDF
    Isoradial graphs are a natural generalization of regular graphs which give, for many models of statistical mechanics, the right framework for studying models at criticality. In this survey paper, we first explain how isoradial graphs naturally arise in two approaches used by physicists: transfer matrices and conformal field theory. This leads us to the fact that isoradial graphs provide a natural setting for discrete complex analysis, to which we dedicate one section. Then, we give an overview of explicit results obtained for different models of statistical mechanics defined on such graphs: the critical dimer model when the underlying graph is bipartite, the 2-dimensional critical Ising model, random walk and spanning trees and the q-state Potts model.Comment: 22 page
    corecore