47 research outputs found
Fast domino tileability
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
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
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
In a region 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 is
defined on the set of all tilings of such that two tilings are adjacent if
we change one to another by a flip (a rotation of a pair of
side-by-side dominoes). It is well-known that is connected
when is simply connected. By using graph theoretical approach, we show that
the flip graph of quadriculated cylinder is still connected,
but the flip graph of quadriculated torus is disconnected and
consists of exactly two isomorphic components. For a tiling , we associate
an integer , forcing number, as the minimum number of dominoes in
that is contained in no other tilings. As an application, we obtain that the
forcing numbers of all tilings in quadriculated cylinder and
torus form respectively an integer interval whose maximum value is
Quadri-tilings of the plane
We introduce {\em quadri-tilings} and show that they are in bijection with
dimer models on a {\em family} of graphs 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 , we prove an
explicit expression, only depending on the local geometry of the graph ,
for the minimal free energy per fundamental domain Gibbs measure; this solves a
conjecture of \cite{Kenyon1}. We also show that when edges of 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
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