A branched polymer of order n in R D —or just “polymer ” for short—is a connected set of n labeled unit spheres with nonoverlapping interiors. We will assume that the sphere labeled 1 is...
These are lecture notes for lectures at the Park City Math Institute, summer 2007. We cover aspects of the dimer model on planar, periodic bipartite graphs, including local statistics, limit shapes...
A branched polymer is a connected configuration of non-overlapping unit balls in space. Building on and from the work of Brydges and Imbrie, we give an elementary calculation of the volume of the...
Monotone loop models and rational resonance (2008)
Hammond, Alan, Kenyon, Richard
Let $T_{n,m}=\mathbb Z_n\times\mathbb Z_m$, and define a random mapping $\phi\colon T_{n,m}\to T_{n,m}$ by $\phi(x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability....
A GROUP OF PATHS IN R 2 (2008)
Abstract. We define a group structure on the set of compact “minimal” paths in R 2. We classify all finitely generated subgroups of this group G: they are free products of free abelian groups and...
1. Dimer problems, Encyclopedia of Mathematical physics, to appear. (2008)
Richard Kenyon, A. Okounkov, Ann Math, Planar Dimers, Harnack Curves, ...
5. What is... a dimer?, with A. Okounkov. Notices Amer. Math. Soc. 52 (2005), no. 3, 342-343.
On the characterization of expansion maps for self-affine tilings (2008)
Kenyon, Richard, Solomyak, Boris
We consider self-affine tilings in $\R^n$ with expansion matrix $\phi$ and address the question which matrices $\phi$ can arise this way. In one dimension, $\lambda$ is an expansion factor of a...
Dimères Et Arbres Couvrants. (2007)
Introduction La m'ecanique statistique tente `a d'ecrire le comportement d'un syst`eme physique ferm'e quand on varie des param`etres ext'erieurs, par exemple la...
Introduction Le probl`eme fondamental de la physique statistique est d"etudier les syst`emes d'un grand nombre d'atomes. Si le syst`eme est suffisamment grand, on peut esp`erer...
Presentation Des Travaux (2007)
Introduction Bien qu'on pense souvent aux pavages comme faisant partie des math'ematiques r'ecr'eatives, nous montrerons ici que la th'eorie des pavages a aussi des aspects...
The low-temperature expansion of the Wulff crystal in the three-dimensional Ising model (2007)
Abstract. We compute the expansion of the surface tension of the 3D random cluster model for q 1 in the limit where p goes to 1. We also compute the asymptotic shape of a plane partition of n as n...
Kenyon, Richard, Winkler, Peter
Building on and from the work of Brydges and Imbrie, we give an elementary calculation of the volume of the space of branched polymers of order $n$ in the plane and in 3-space. Our development...
Kenyon, Richard, Okounkov, Andrei, Sheffield, Scott
We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph $G$ embedded in the plane. We derive...
Planar dimers and Harnack curves (2006)
Kenyon, Richard, Okounkov, Andrei
In this article we study the connection between dimers and Harnack curves discovered in [15]. We prove that every Harnack curve arises as a spectral curve of some dimer model. We also prove that the...
Kenyon, Richard, Sheffield, Scott, Okounkov, Andrei
We study random surfaces which arise as height functions of random perfect matchings (a.k.a. dimer configurations) on a weighted, bipartite, doubly periodic graph G embedded in the plane. We derive...
Limit shapes and the complex burgers equation (2005)
Kenyon, Richard, Okounkov, Andrei
In this paper we study surfaces in R^3 that arise as limit shapes in a class of random surface models arising from dimer models. The limit shapes are minimizers of a surface tension functional, that...
Topological mixing for substitutions on two letters (2004)
Kenyon, Richard, Sadun, Lorenzo, Solomyak, Boris
We investigate topological mixing for Z and R actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $\theta_2$ of the substitution...
Height fluctuations in the honeycomb dimer model (2004)
We study a model of random surfaces arising in the dimer model on the honeycomb lattice. For a fixed ``wire frame'' boundary condition, as the lattice spacing $\epsilon\to0$, Cohn, Kenyon and Propp...
Kenyon, Richard, Okounkov, Andrei, Sheffield, Scott
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...
Planar dimers and Harnack curves (2003)
Kenyon, Richard, Okounkov, Andrei
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...
An introduction to the dimer model (2003)
Lecture notes from a minicourse given at the ICTP in May 2002.
Dimers, Tilings and Trees (2003)
Kenyon, Richard, Sheffield, Scott
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;...
Rhombic embeddings of planar graphs with faces of degree 4 (2003)
Kenyon, Richard, Schlenker, Jean-Marc
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...
The Laplacian and $\bar\partial$ operators on critical planar graphs (2002)
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...
Coagulation and Complement Studies in Rocky Mountain Spotted Fever, (2002)
Fine,Douglas, Mosher,Deane, Yamada,Tadataka, Burke,Donald, Kenyon,Richard
Dominos and the Gaussian Free Field (2001)
We define a scaling limit of the height function on the domino tiling model (dimer model) on simply connected regions in $\mathbf{Z}^2$ and show that it is the “massless free field,” a Gaussian...
Local statistics of lattice dimers (2001)
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...
The asymptotic determinant of the discrete Laplacian (2000)
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...
A variational principle for domino tilings (2000)
Cohn, Henry, Kenyon, Richard, Propp, James
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...
Conformal invariance of domino tiling (2000)
Let $U$ be a multiply connected region in $\mathbf{R}^ 2$ with smooth boundary. Let $P_\epsilon$ be a polyomino in $\epsilon\mathbf{Z}^2$ approximating $U$ as $\epsilon \to 0$.We show that, for...
Conformal invariance of domino tiling (1999)
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...
Conformal Invariance of Domino Tiling (1999)
this paper we deal with the two-dimensional lattice dimer model, or domino tiling model (a domino tiling is a tiling with 2 \Theta 1 and 1 \Theta 2 rectangles). We prove that in the limit as the...
Loop-Erased Random Walks (1999)
Introduction The loop-erased random walk is the simple curve obtained by removing in the chronological order the loops of the original random walk. Two aspects of these walks in Z 2 are studied: its...
The asymptotic determinant of the discrete Laplacian (1999)
We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R 2 . Specifically, for each ffl ? 0 let H ffl be the subgraph of fflZ 2 whose vertices lie...
A variational principle for domino tilings (1998)
Henry Cohn, Richard Kenyon, James Propp, Dedicated Pieter, Willem Kasteleyn
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...
Billiards on Rational-Angled Triangles (1998)
this paper. The lattice polygons are those polygons for which the surface M ` has a large affine automorphism group (see section 3). In this paper we introduce techniques to analyze affine...
The Planar Dimer Model With Boundary: A Survey. (1998)
this paper we would like to give a short survey of some of these new results.
A Variational Principle For Domino Tilings (1998)
Henry Cohn, Richard Kenyon, James Propp, Dedicated Pieter, Willem Kasteleyn
. 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...
Geometry Of Self-Affine Tiles II (1998)
Richard Kenyon, Jie Li, Robert S. Strichartz, Yang Wang
. We continue the study in part I of geometric properties of self--similar and self--affine tiles. We give some experimental results from implementing the algorithm in part I for computing the...
Arithmetic Construction of Sofic Partitions of Hyperbolic Toral Automorphisms (1998)
Richard Kenyon, Anatoly Vershik
For each irreducible hyperbolic automorphism A of the n-torus we construct a sofic system (\Sigma; oe) and an isomorphism of (\Sigma; oe) with a finite cover of (T n ; A). This construction is...
A variational principle for domino tilings (1998)
Henry Cohn, Richard Kenyon, James Propp, Dedicated Pieter, Willem Kasteleyn
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.
Local Statistics Of Lattice Dimers (1997)
. 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...
James Cannon William, William J. Floyd, Richard Kenyon, R. Parry
Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6....
James W. Cannon, William J. Floyd, Richard Kenyon, R. Parry
3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65
A note on tiling with integer-sided rectangles (1996)
We show how to determine if a given simple rectilinear polygon can be tiled with rectangles, each having an integer side.
A Note on Tiling With Integer-Sided Rectangles. (1996)
We show how to determine if a given simple rectilinear polygon can be tiled with rectangles, each having an integer side. 1 Introduction In [6], Stan Wagon provides us with 14 proofs of the fact that...
The Construction of Self-Similar Tilings (1995)
We give a construction of a self-similar tiling of the plane with any prescribed expansion coefficient $\lambda\in\C$ (satisfying the necessary algebraic condition of being a complex Perron number)....
Tiling a rectangle with the fewest squares (1994)
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...
A note on tiling with integer-sided rectangles (1994)
We show how to determine if a given simple rectilinear polygon can be tiled with rectangles, each having an integer side.
Tiling a Polygon with Rectangles (1992)
We study the problem of tiling a simple polygon of surface n with rectangles of given types (tiles). We present a linear time algorithm for deciding if a polygon can be tiled with 1 \Theta m and k...
Tiling With Squares and Square-Tileable Surfaces.
We introduce the `square tiling group' and use it to give necessary conditions on a planar polygon to be tileable with squares. We define square tilings on Riemann surfaces, and compute the...
Long-Range Properties of Spanning Trees
We compute some large-scale properties of the uniform spanning tree process on Z 2 . In particular we compute certain crossing probabilities for rectangles and annuli. 1 Introduction On a finite...