arXiv subjects

Richard Kenyon

Publications and source records attributed to Richard Kenyon.

At least 55 records · Page 3Linked to original sources

arXiv2015

Bipodal structure in oversaturated random graphs

We study the asymptotics of large simple graphs constrained by the limiting density of edges and the limiting subgraph density of an arbitrary fixed graph $H$. We prove that, for all but finitely many values of the edge density, if the density of $H$ is constrained to be slightly higher than that for the corresponding Erdős-Rényi graph, the typical large graph is bipodal with parameters varying analytically with the densities. Asymptotically, the parameters depend only on the degree sequence of $H$.

math.CO↗

arXiv2015

Permutations with fixed pattern densities

We study scaling limits of random permutations ("permutons") constrained by having fixed densities of a finite number of patterns. We show that the limit shapes are determined by maximizing entropy over permutons with those constraints. In particular, we compute (exactly or numerically) the limit shapes with fixed \hbox{12} density, with fixed \hbox{12} and \hbox{123} densities, with fixed \hbox{12} density and the sum of \hbox{123} and \hbox{213} densities, and with fixed \hbox{123} and \hbox{321} densities. In the last case we explore a particular phase transition. To obtain our results, we also provide a description of permutons using a dynamic construction.

math.CO↗

arXiv2015

Right-angled hexagon tilings of the hyperbolic plane

We study isometry-invariant probability measures on the space $Ω$ of tilings of the hyperbolic plane with right-angled hexagons of varying shapes. We prove that, for each measure $μ$ in a certain natural family of measures on right-angled hexagons, there is an isometry-invariant measure on $Ω$ whose marginal distribution on tiles is $μ$.

math.PR↗

arXiv2014

Principal minors and rhombus tilings

The algebraic relations between the principal minors of an $n\times n$ matrix are somewhat mysterious, see e.g. [lin-sturmfels]. We show, however, that by adding in certain \emph{almost} principal minors, the relations are generated by a single relation, the so-called hexahedron relation, which is a composition of six cluster mutations. We give in particular a Laurent-polynomial parameterization of the space of $n\times n$ matrices, whose parameters consist of certain principal and almost principal minors. The parameters naturally live on vertices and faces of the tiles in a rhombus tiling of a convex $2n$-gon. A matrix is associated to an equivalence class of tilings, all related to each other by Yang-Baxter-like transformations. By specializing the initial data we can similarly parametrize the space of Hermitian symmetric matrices over $\mathbb R, \mathbb C$ or $\mathbb H$ the quaternions. Moreover by further specialization we can parametrize the space of \emph{positive definite} matrices over these rings.

math.CO↗

arXiv2014

Random two-component spanning forests

We study random two-component spanning forests ($2$SFs) of finite graphs, giving formulas for the first and second moments of the sizes of the components, vertex-inclusion probabilities for one or two vertices, and the probability that an edge separates the components. We compute the limit of these quantities when the graph tends to an infinite periodic graph in ${\mathbb R}^d$.

math.PR↗

arXiv2013

Double-dimers, the Ising model and the hexahedron recurrence

We define and study a recurrence relation in ${\mathbb Z}^3$, called the hexahedron recurrence, which is similar to the octahedron recurrence (Hirota bilinear difference equation) and cube recurrence (Miwa equation). Like these examples, solutions to the hexahedron recurrence are partition sums for edge configurations on a certain graph, and have a natural interpretation in terms of cluster algebras. We give an explicit correspondence between monomials in the Laurent expansions arising in the recurrence with certain double-dimer configurations of a graph. We compute limit shapes for the corresponding double-dimer configurations. The Kashaev difference equation arising in the Ising model star-triangle relation is a special case of the hexahedron recurrence. In particular this reveals the cluster nature underlying the Ising model. The above relation allows us to prove a Laurent phenomenon for the Kashaev difference equation.

math-ph↗

arXiv2012

Conformal invariance of loops in the double-dimer model

The dimer model is the study of random dimer covers (perfect matchings) of a graph. A double-dimer configuration on a graph $G$ is a union of two dimer covers of $G$. We introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes (relative to a fixed set of faces) of loops in the double dimer model on a planar graph. As an application we prove that, in the scaling limit of the "uniform" double-dimer model on ${\mathbb Z}^2$ (or on any other bipartite planar graph conformally approximating $\mathbb C$), the loops are conformally invariant. As other applications we compute the exact distribution of the number of topologically nontrivial loops in the double-dimer model on a cylinder and the expected number of loops surrounding two faces of a planar graph.

math.PR↗

arXiv2012

The Laplacian on planar graphs and graphs on surfaces

These are lecture notes for the Current Developments in Mathematics conference at Harvard, November, 2011. We discuss topological, probabilistic and combinatorial aspects of the Laplacian on a graph embedded on a surface. The three main goals are to discuss: (1) for "circular" planar networks, the characterization due to Colin de Verdière of Dirichlet-to-Neumann operator; (2) The connections with the random spanning tree model; and (3) the characteristic polynomial of the Laplacian on an annulus and torus.

math.PR↗

arXiv2011

Spanning forests and the vector bundle Laplacian

The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests (CRSFs). We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example, we compute the probability that a loop-erased random walk on a planar graph between two vertices on the outer boundary passes left of two given faces. This probability cannot be computed using the standard Laplacian alone.

math.PR↗

arXiv2011

Hausdorff dimension for fractals invariant under the multiplicative integers

We consider subsets of the (symbolic) sequence space that are invariant under the action of the semigroup of multiplicative integers. A representative example is the collection of all 0-1 sequences $(x_k)$ such that $x_k x_{2k}=0$ for all $k$. We compute the Hausdorff and Minkowski dimensions of these sets and show that they are typically different. The proof proceeds via a variational principle for multiplicative subshifts.

math.DS↗

arXiv2011

Hausdorff dimension of the multiplicative golden mean shift

We compute the Hausdorff dimension of the "multiplicative golden mean shift" defined as the set of all reals in $[0,1]$ whose binary expansion $(x_k)$ satisfies $x_k x_{2k}=0$ for all $k\ge 1$, and show that it is smaller than the Minkowski dimension.

math.DS↗

arXiv2010

On the characterization of expansion maps for self-affine tilings

We consider self-affine tilings in $\R^n$ with expansion matrix $ϕ$ and address the question which matrices $ϕ$ can arise this way. In one dimension, $λ$ is an expansion factor of a self-affine tiling if and only if $|λ|$ is a Perron number, by a result of Lind. In two dimensions, when $ϕ$ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex $λ$ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for $ϕ$ to be an expansion matrix for any $n$, assuming only that $ϕ$ is diagonalizable over the complex numbers. We conjecture that this condition on $ϕ$ is also sufficient for the existence of a self-affine tiling.

math.MG↗

arXiv2009

Lectures on Dimers

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 and fluctuations.

math.PR↗

arXiv2008

Monotone loop models and rational resonance

Let $T_{n,m}=\mathbb Z_n\times\mathbb Z_m$, and define a random mapping $ϕ\colon T_{n,m}\to T_{n,m}$ by $ϕ(x,y)=(x+1,y)$ or $(x,y+1)$ independently over $x$ and $y$ and with equal probability. We study the orbit structure of such ``quenched random walks'' $ϕ$ in the limit $m,n\to\infty$, and show how it depends sensitively on the ratio $m/n$. For $m/n$ near a rational $p/q$, we show that there are likely to be on the order of $\sqrt{n}$ cycles, each of length O(n), whereas for $m/n$ far from any rational with small denominator, there are a bounded number of cycles, and for typical $m/n$ each cycle has length on the order of $n^{4/3}$.

math.PR↗

arXiv2007

Branched Polymers

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 reveals some more general identities, and allows exact random sampling. In particular we show that a random 3-dimensional branched polymer of order $n$ has diameter of order $\sqrt{n}$.

math.PR↗

arXiv2007

Height fluctuations in the honeycomb dimer model

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 $ε\to0$, Cohn, Kenyon and Propp [CKP] showed the almost sure convergence of a random surface to a non-random limit shape $Σ_0$. In [KO], Okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions, in particular constructing a natural conformal structure on them. We show here that when $Σ_0$ has no facets, for a family of boundary conditions approximating the wire frame, the large-scale surface fluctuations (height fluctuations) about $Σ_0$ converge as $ε\to0$ to a Gaussian free field for the above conformal structure. We also show that the local statistics of the fluctuations near a given point $x$ are, as conjectured in [CKP], given by the unique ergodic Gibbs measure (on plane configurations) whose slope is the slope of the tangent plane of $Σ_0$ at $x$.

math-ph↗

arXiv2007

Limit shapes and the complex burgers equation

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 is, they minimize, for fixed boundary conditions, the integral of a quantity (the surface tension) depending only on the slope of the surface. The surface tension as a function of the slope has singularities and is not strictly convex, which leads to formation of facets and edges in the limit shapes. We find a change of variables that reduces the Euler-Lagrange equation for the variational problem to the complex inviscid Burgers equation (complex Hopf equation). The equation can thus be solved in terms of an arbitrary holomorphic function, which is somewhat similar in spirit to Weierstrass parametrization of minimal surfaces. We further show that for a natural dense set of boundary conditions, the holomorphic function in question is, in fact, algebraic. The tools of algebraic geometry can thus be brought in to study the the minimizers and, especially, the formation of their singularities. This is illustrated by several explicitly computed examples.

math-ph↗

arXiv2005

Topological mixing for substitutions on two letters

We investigate topological mixing for Z and R actions associated with primitive substitutions on two letters. The characterization is complete if the second eigenvalue $θ_2$ of the substitution matrix satisfies $|θ_2|\ne 1$. If $|θ_2|<1$, then (as is well-known) the substitution system is not topologically weak mixing, so it is not topologically mixing. We prove that if $|θ_2|> 1$, then topological mixing is equivalent to topological weak mixing, which has an explicit arithmetic characterization. The case $|θ_2|=1$ is more delicate, and we only obtain some partial results.

math.DS↗