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Richard Kenyon

Publications and source records attributed to Richard Kenyon.

67 records · Page 4Linked to original sources
arXiv2004

Planar dimers and Harnack curves

In this paper we study the connection between dimers and Harnack curves discovered in math-ph/0311005. We prove that every Harnack curve arises as a spectral curve of some dimer model. We also prove that the space of Harnack curve of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates. We characterize Harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their Ronkin function with given boundary conditions.

math.AG↗
arXiv2003

Dimers and Amoebae

We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on an weighted, bipartite, doubly periodic graph G embedded in the plane. We derive explicit formulas for the surface tension and local Gibbs measure probabilities of these models. The answers involve a certain plane algebraic curve, which is the spectral curve of the Kasteleyn operator of the graph. For example, the surface tension is the Legendre dual of the Ronkin function of the spectral curve. The amoeba of the spectral curve represents the phase diagram of the dimer model. Further, we prove that the spectral curve of a dimer model is always a real curve of special type, namely it is a Harnack curve. This implies many qualitative and quantitative statement about the behavior of the dimer model, such as existence of smooth phases, decay rate of correlations, growth rate of height function fluctuations, etc.

math-ph↗
arXiv2003

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others 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 on 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.

math.CO↗
arXiv2003

Rhombic embeddings of planar graphs with faces of degree 4

Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings.

math-ph↗
arXiv2002

The Laplacian and $\bar\partial$ operators on critical planar graphs

On a periodic planar graph whose edge weights satisfy a certain simple geometric condition, the discrete Laplacian and d-bar operators have the property that their determinants and inverses only depend on the local geometry of the graph. We obtain explicit expressions for the logarithms of the (normalized) determinants, as well as the inverses of these operators. We relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron. In the associated dimer and spanning tree models, for which the determinants of d-bar and the Laplacian respectively play the role of the partition function, this allows us to compute the entropy and correlations in terms of the local geometry. In addition, we define a continuous family of special discrete analytic functions, which, via convolutions gives a general process for constructing discrete analytic functions and discrete harmonic functions on critical planar graphs.

math-ph↗
arXiv2001

A variational principle for domino tilings

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 epsilon (for an appropriate metric) of the unique entropy-maximizing 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.

math.CO↗
arXiv2001

Local statistics of lattice dimers

We show how to compute the probability of any given local configuration in a random tiling of the plane with dominos. That is, we explicitly compute the measures of cylinder sets for the measure of maximal entropy $μ$ on the space of tilings of the plane with dominos. We construct a measure $ν$ on the set of lozenge tilings of the plane, show that its entropy is the topological entropy, and compute explicitly the $ν$-measures of cylinder sets. As applications of these results, we prove that the translation action is strongly mixing for $μ$ and $ν$, and compute the rate of convergence to mixing (the correlation between distant events). For the measure $ν$ we compute the variance of the height function.

math.CO↗
arXiv2000

The asymptotic determinant of the discrete Laplacian

We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R^2. As an application of this result, we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.

math-ph↗
arXiv1999

Conformal invariance of domino tiling

Let U be a multiply-connected region in R^2 with smooth boundary. Let P_epsilon be a polyomino in epsilon Z^2 approximating U as epsilon tends to 0. We show that, for certain boundary conditions on P_epsilon, the height distribution on a random domino tiling (dimer covering) of P_epsilon is conformally invariant in the limit as epsilon tends to 0, in the sense that the distribution of heights of boundary components only depends on the conformal type of U. The mean height and all the moments are explicitly evaluated.

math-ph↗
arXiv1995

The Construction of Self-Similar Tilings

We give a construction of a self-similar tiling of the plane with any prescribed expansion coefficient $λ\in\C$ (satisfying the necessary algebraic condition of being a complex Perron number). For any integer $m>1$ we show that there exists a self-similar tiling with $2π/m$-rotational symmetry group and expansion $λ$ if and only if either $λ$ or $λe^{2πi/m}$ is a complex Perron number for which $e^{2πi/m}$ is in $\Q[λ]$, respectively $Q[λe^{2πi/m}]$.

math.MG↗
arXiv1994

Tiling a rectangle with the fewest squares

We show that a square-tiling of a $p\times q$ rectangle, where $p$ and $q$ are relatively prime integers, has at least $\log_2p$ squares. If $q>p$ we construct a square-tiling with less than $q/p+C\log p$ squares of integer size, for some universal constant $C$.

math.CO↗