Results 1  10
of
21
Correlation function of Schur process with application to local geometry of a random 3dimensional Young Diagram
, 2001
"... ..."
A variational principle for domino tilings
"... Abstract. 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 entrop ..."
Abstract

Cited by 102 (15 self)
 Add to MetaCart
(Show Context)
Abstract. 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 ε (for an appropriate metric) of the unique entropymaximizing 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. The effect of boundary conditions is, however, not entirely trivial and will be discussed in more detail in a subsequent paper. P. W. Kasteleyn, 1961 1.
Thermodynamic limit of the sixvertex model with domain wall boundary conditions
"... We address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model. Here we compare the bulk free energy for periodic and domain wall boundary conditions. Using a determinant representation for the partition function with domain wall boundary conditions ..."
Abstract

Cited by 40 (6 self)
 Add to MetaCart
(Show Context)
We address the question of the dependence of the bulk free energy on boundary conditions for the six vertex model. Here we compare the bulk free energy for periodic and domain wall boundary conditions. Using a determinant representation for the partition function with domain wall boundary conditions, we derive Toda differential equations and solve them asymptotically in order to extract the bulk free energy. We find that it is different and bears no simple relation with the free energy for periodic boundary conditions. The six vertex model with domain wall boundary conditions is closely related to algebraic combinatorics (alternating sign matrices). This implies new results for the weighted counting for large size alternating sign matrices. Finally we comment on the interpretation of our results, in particular in connection with domino tilings (dimers on a square lattice). 04/2000
Diffraction of Random Tilings: Some Rigorous Results
 J. STAT. PHYS
, 1999
"... The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar rando ..."
Abstract

Cited by 30 (17 self)
 Add to MetaCart
The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of stochastic product tilings built from cuboids, and of planar random tilings based on solvable dimer models, augmented by a brief outline of the diraction from the classical 2D Ising lattice gas. We also give a summary of the measure theoretic approach to mathematical diraction theory which underlies the unique decomposition of the diffraction spectrum into its pure point, singular continuous and absolutely continuous parts.
SixVertex Model with Domain Wall Boundary Conditions and OneMatrix Model
, 2000
"... The partition function of the sixvertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a hermitean onematrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, w ..."
Abstract

Cited by 29 (1 self)
 Add to MetaCart
The partition function of the sixvertex model on a square lattice with domain wall boundary conditions (DWBC) is rewritten as a hermitean onematrix model or a discretized version of it (similar to sums over Young diagrams), depending on the phase. The expression is exact for finite lattice size, which is equal to the size of the corresponding matrix. In the thermodynamic limit, the matrix integral is computed using traditional matrix model techniques, thus providing a complete treatment of the bulk free energy of the sixvertex model with DWBC in the different phases. In particular, in the antiferroelectric phase, the bulk free energy and a subdominant correction are given exactly in terms of elliptic theta functions.
Random Surfaces
, 2006
"... We study the statistical physical properties of (discretized) “random surfaces, ” which are random functions from Z d (or large subsets of Z d) to E, where E is Z or R. Their laws are determined by convex, nearestneighbor, gradient Gibbs potentials that are invariant under translation by a fullran ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
We study the statistical physical properties of (discretized) “random surfaces, ” which are random functions from Z d (or large subsets of Z d) to E, where E is Z or R. Their laws are determined by convex, nearestneighbor, gradient Gibbs potentials that are invariant under translation by a fullrank sublattice L of Z d; they include many discrete and continuous height function models (e.g., domino tilings, square ice, the harmonic crystal, the GinzburgLandau ∇φ interface model, the linear solidonsolid model) as special cases. We prove a variational principle—characterizing gradient phases of a given slope as minimizers of the specific free energy—and an empirical measure large deviations principle (with a unique rate function minimizer) for random surfaces on mesh approximations of bounded domains. We also prove that the surface tension is strictly convex and that if u is in the interior of the space of finitesurfacetension slopes, then there exists a minimal energy gradient phase µu of slope u. Using a new geometric technique called cluster swapping (a variant of the SwendsenWang update for FortuinKasteleyn clusters), we show that µu is unique if at least one of the following holds: E = R, d ∈ {1, 2}, there exists a rough gradient phase of slope u, or u is irrational. When d = 2 and E = Z, we show that the slopes of all smooth phases (a.k.a. crystal facets) lie in the dual lattice of L. In the case E = Z and d = 2, our results resolve and greatly generalize a number of conjectures of Cohn, Elkies, and Propp—one of which is that there is a unique ergodic Gibbs measure on domino tilings for each nonextremal slope. We also prove several theorems cited by Kenyon, Okounkov, and Sheffield in their recent exact solution of the dimer model on general planar lattices. In the case E = R, our results generalize and extend many of the results in the literature on GinzurgLandau ∇φinterface models.
Step fluctuations for a faceted crystal
, 2008
"... A statistical mechanics model for a faceted crystal is the 3D Ising model at zero temperature. It is assumed that in one octant all sites are occupied by atoms, the remaining ones being empty. Allowed atom configurations are such that they can be obtained from the filled octant through successive re ..."
Abstract

Cited by 20 (3 self)
 Add to MetaCart
(Show Context)
A statistical mechanics model for a faceted crystal is the 3D Ising model at zero temperature. It is assumed that in one octant all sites are occupied by atoms, the remaining ones being empty. Allowed atom configurations are such that they can be obtained from the filled octant through successive removals of atoms with breaking of precisely three bonds. If V denotes the number of atoms removed, then the grand canonical Boltzmann weight is q V, 0 < q < 1. As shown by Cerf and Kenyon, in the limit q → 1 a deterministic shape is attained, which has the three facets (100), (010), (001), and a rounded piece interpolating between them. We analyse the step statistics as q → 1. In the rounded piece it is given by a determinantal process based on the discrete sinekernel. Exactly at the facet edge, the steps have more space to meander. Their statistics is again determinantal, but this time based on the Airykernel. In particular, the border step is well approximated by the Airy process, which has been obtained previously in the context of growth models. Our results are based on the asymptotic analysis for spacetime inhomogeneous transfer matrices.
Constrained Codes as Networks of Relations
"... Abstract — We address the wellknown problem of determining the capacity of constrained coding systems. While the onedimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a twodimensional ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
(Show Context)
Abstract — We address the wellknown problem of determining the capacity of constrained coding systems. While the onedimensional case is well understood to the extent that there are techniques for rigorously deriving the exact capacity, in contrast, computing the exact capacity of a twodimensional constrained coding system is still an elusive research challenge. The only known exception in the twodimensional case is an exact (however, not rigorous) solution to the (1, ∞)RLL system on the hexagonal lattice. Furthermore, only exponentialtime algorithms are known for the related problem of counting the exact number of constrained twodimensional information arrays. We present the first known rigorous technique that yields an exact capacity of a twodimensional constrained coding system. In addition, we devise an efficient (polynomial time) algorithm for counting the exact number of constrained arrays of any given size. Our approach is a composition of a number of ideas and techniques: describing the capacity problem as a solution to a counting problem in networks of relations, graphtheoretic tools originally developed in the field of statistical mechanics, techniques for efficiently simulating quantum circuits, as well as ideas from the theory related to the spectral distribution of Toeplitz matrices. Using our technique we derive a closed form solution to the capacity related to the PathCover constraint in a twodimensional triangular array (the resulting calculated capacity is 0.72399217...). PathCover is a generalization of the well known onedimensional (0, 1)RLL constraint for which the capacity is known to be 0.69424... Index Terms — capacity of constrained systems, capacity of twodimensional constrained systems, holographic reductions, networks of relations, FKT method, spectral distribution of Toeplitz matrices I.
Dimers on twodimensional lattices
, 2008
"... We consider closepacked dimers, or perfect matchings, on twodimensional regular lattices. We review known results and derive new expressions for the free energy, entropy, and the molecular freedom of dimers for a number of lattices including the simplequartic (4 4), honeycomb (6 3), triangular (3 ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
We consider closepacked dimers, or perfect matchings, on twodimensional regular lattices. We review known results and derive new expressions for the free energy, entropy, and the molecular freedom of dimers for a number of lattices including the simplequartic (4 4), honeycomb (6 3), triangular (3 6), kagomé (3 · 6 · 3 · 6), 312 (3 · 12 2) and its dual [3 · 12 2], and 48 (4 · 8 2) and its dual Union Jack [4 · 8 2] Archimedean tilings. The occurrence and nature of phase transitions are also discussed. Key words: Closepacked dimers, twodimensional lattices, exact results, phase transitions. 1 2 1