Related papers: Tilings With Very Elastic Dimers
Quantum many-body states that frequently appear in physics often obey an entropy scaling law, meaning that an entanglement entropy of a subsystem can be expressed as a sum of terms that scale linearly with its volume and area, plus a…
Domino tilings have been studied extensively for both their statistical properties and their dynamical properties. We construct a subshift of finite type using matching rules for several types of dominos. We combine the previous results…
We consider substitution tilings and Delone sets without the assumption of finite local complexity (FLC). We first give a sufficient condition for tiling dynamical systems to be uniquely ergodic and a formula for the measure of cylinder…
Penrose tilings form lattices, exhibiting 5-fold symmetry and isotropic elasticity, with inhomogeneous coordination much like that of the force networks in jammed systems. Under periodic boundary conditions, their average coordination is…
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union…
We first prove that the set of domino tilings of a fixed finite figure is a distributive lattice, even in the case when the figure has holes. We then give a geometrical interpretation of the order given by this lattice, using (not…
Many glass-forming fluids exhibit a remarkable thermodynamic scaling in which dynamic properties, such as the viscosity, the relaxation time, and the diffusion constant, can be described under different thermodynamic conditions in terms of…
We show that the problem of tiling the Euclidean plane with a finite set of polygons (up to translation) boils down to prove the existence of zeros of a non-negative convex function defined on a finite-dimensional simplex. This function is…
We give a complete rigorous proof of the full asymptotic expansion of the partition function of the dimer model on a square lattice on a torus for general weights $z_h, z_v$ of the dimer model and arbitrary dimensions of the lattice $m, n$.…
Let G be a finite graph or an infinite graph on which Z^d acts with finite fundamental domain. If G is finite, let T be a random spanning tree chosen uniformly from all spanning trees of G; if G is infinite, known methods show that this…
We study here slopes of periodicity of tilings. A tiling is of slope if it is periodic along direction but has no other direction of periodicity. We characterize in this paper the set of slopes we can achieve with tilings, and prove they…
The problem that we consider is the following: given an $n \times n$ array $A$ of positive numbers, find a tiling using at most $p$ rectangles (which means that each array element must be covered by some rectangle and no two rectangles must…
We consider a randomly forced Ginzburg-Landau equation on an unbounded domain. The forcing is smooth and homogeneous in space and white noise in time. We prove existence and smoothness of solutions, existence of an invariant measure for the…
We introduce a new geometric approach to a manifold equipped with a smooth density function that takes a torsion-free affine connection, as opposed to a weighted measure or Laplacian, as the fundamental object of study. The connection…
Entropy is a quantity for counting physical degrees of freedom in a system. At a finite temperature, one can use thermal entropy to study thermodynamical properties. At zero temperature, entanglement entropy is expected to provide a…
We consider three-dimensional domino tilings of cylinders $\mathcal{D} \times [0,N] \subset \mathbb{R}^3$, where $\mathcal{D} \subset \mathbb{R}^2$ is a balanced quadriculated disk and $N \in \mathbb{N}$. A flip is a local move in the space…
We work with the following expression for the entropy (density) of a dimer gas on an infinite r-regular lattice lambda(p) = 1/2 [ pln(r)-ln(p)-2(1-p)ln(1-p)-p ]+sum_{k=2}(d_k)(p^k) where the indicated sum converges for density, p, small…
It is proved that whenever two aperiodic repetitive tilings with finite local complexity have homeomorphic tiling spaces, their associated complexity functions are asymptotically equivalent in a certain sense (which implies, if the…
Two-dimensional random tilings of rhombi can be seen as projections of two-dimensional membranes embedded in hypercubic lattices of higher dimensional spaces. Here, we consider tilings projected from a $D$-dimensional space. We study the…
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which…