arXiv subjects
Publications and source records attributed to Richard Kenyon.
We derive a determinantal expression for the inhomogeneous asymmetric five vertex model in a rectangular geometry with arbitrary boundary conditions at the bottom and top. Standard non-intersecting lattice path, or free fermion, approaches are not applicable and the determinantal form thus is not immediate.
We study pattern densities in binary sequences, finding optimal limit sequences with fixed pattern densities.
We study a quantum version of the $n$-dimer model from statistical mechanics, based on the formalism from quantum topology developed by Reshetikhin and Turaev (the latter which, in particular, can be used to construct the Jones polynomial of a knot in $\mathbb{R}^3$). We apply this machinery to construct an isotopy invariant polynomial for knotted bipartite ribbon graphs in $\mathbb{R}^3$, giving, in the planar setting, a quantum $n$-dimer partition function. As one application, we compute the expected number of loops in the (classical) double dimer model for planar bipartite graphs.
The dimer model is a classical statistical mechanics model which is exactly solvable in two dimensions, but about which little is known in higher dimensions. In analogy with large $N$ limits in lattice gauge theory, we study a large $N$ limit of the dimer model in any dimension $d$. The dependence on $N$ comes from the multinomial tiling model introduced by Kenyon and Pohoata, which gives a general framework for adding a dependence on $N$ to a tiling model. We study the behavior of this model on periodic bipartite graphs in ${\mathbb R}^d$, in the scaling limit as the multiplicity $N$ and then the size of the graph go to infinity. In this iterated limit, in any dimension $d$, we prove a variational principle and show that random configurations concentrate on a limit shape which is the unique solution to an associated system of Euler-Lagrange equations. The rate function of the variational principle is the integral of a surface tension function, which we can compute explicitly for lattices in any dimension $d$ as the Legendre dual of the free energy for the model on the torus. We give a unified methodology for computing the surface tension and Euler-Lagrange equations in any dimension $d$. A new structure called the critical gauge also emerges in the large $N$ limit. We show that the critical gauge functions converges in the scaling limit to a limiting gauge function which is the unique solution to a dual Euler-Lagrange equation. This limiting gauge function determines the limit shape and vice versa. We further use our techniques to compute explicit limit shapes in some two and three dimensional examples, such as the Aztec diamond and "Aztec cuboid". This is one of the first stat mech models in dimensions $d\ge3$ where limit shapes can be computed explicitly.
We define multideterminantal probability measures, a family of probability measures on $[k]^n$ where $[k]=\{1,2,\dots,k\}$, generalizing determinantal measures (which correspond to the case $k=2$). We give examples coming from the positive Grassmannian, from the dimer model and from the spanning tree model. We characterize kernels of \emph{pure} $k$-determinantal measures as those arising from $k$-tuples of Grassmannian elements whose maximal minors have certain sign restrictions. As a special case we construct all kernels of pure determinantal measures via a pair of elements of $Gr_{n_1,n}$ having corresponding Pl\"ucker coordinates of the same signs. We also define and completely characterize determinantal probability measures on the permutation group $S_n$.
We define $2n$-multiwebs on planar graphs and discuss their relation with $\mathrm{Sp}(2n)$-webs. On a planar graph with a symplectic local system we define a matrix whose Pfaffian is the sum of traces of $2n$-multiwebs. As application we generalize Kasteleyn's theorem from dimer covers to $2n$-multiweb covers of planar graphs with $U(n)$ gauge group. For $\mathrm{Sp}(4)$ we relate Kuperberg's ``tetravalent vertex'' to the determinant, and classify reduced $4$-webs on some simple surfaces: the annulus, torus, and pair of pants. We likewise define, for $\mathrm{Sp}(2n)$ and $q=1$, a $2n$-valent vertex corresponding to the determinant, and classify reduced $2n$-webs on an annulus.
We study limit shapes in two equivalent models: the six-vertex model in the $c\to0$ limit and the random Mallows permutation with restricted permutation matrix. We give the Euler-Lagrange equation for the limit shape and show how to solve it for a class of rectilinear polygonal domains. Its solutions are given by piecewise-algebraic functions with lines of discontinuities.
We study pairs of matrices $A,B\in GL_n({\mathbb C})$ such that the eigenvalues of $A$, of $B$ and of the product $AB$ are specified in advance. We show that the space of such pairs $(A,B)$ under simultaneous conjugation has dimension $(n-1)(n-2)$, and give an explicit parameterization. More generally let $\Sigma$ be a surface of genus $g$ with $k$ punctures. We find a parameterization of the space $\Omega_{g,k,n}$ of flat $GL_n({\mathbb C})$-structures on $\Sigma$ whose holonomies around the punctures have prescribed eigenvalues. We show furthermore that, for $3\le k\le 2g+6$ (or $3\le k\le 9$ if $g=1$, or $3\le k$ if $g=0$), the space $\Omega_{g,k,n}$ has an explicit symplectic structure and an associated Liouville integrable system, equivalent to a leaf of a Goncharov-Kenyon dimer integrable system.
For partially ordered sets $X$ we consider the square matrices $M^{X}$ with rows and columns indexed by linear extensions of the partial order on $X$. Each entry $\left( M^{X}\right)_{PQ}$ is a formal variable defined by a pedestal of the linear order $Q$ with respect to linear order $P$. We show that all the eigenvalues of any such matrix $M^{X}$ are $\mathbb{Z}$-linear combinations of those variables.
We discuss some diverse open problems in the dimer model, motivated by a geometric viewpoint. This is part of a conference proceedings for the OPAC 2022 conference.
We compute connection probabilities for reduced $3$-webs in the triple-dimer model on circular planar graphs using the boundary measurement matrix (reduced Kasteleyn matrix). As one application we compute several "$\text{SL}_3$ generalizations'' of the Lindstr{\o}m-Gessel-Viennot theorem, for "parallel" webs and for honeycomb webs. We also apply our results to the scaling limit of the dimer model in a planar domain, giving conformally invariant expressions for reduced web probabilities.
Let $G$ be a bipartite planar graph with edges directed from black to white. For each vertex $v$ let $n_v$ be a positive integer. A multiweb in $G$ is a multigraph with multiplicity $n_v$ at vertex $v$. A connection is a choice of linear maps on edges $\Phi=\{\phi_{bw}\}_{bw\in E}$ where $\phi_{bw}\in \mathrm{Hom}({\mathbb R}^{n_b},{\mathbb R}^{n_w})$. Associated to $\Phi$ is a function on multiwebs, the trace $Tr_{\Phi}$. We define an associated Kasteleyn matrix $K=K(\Phi)$ in this setting and write $\det K$ as the sum of traces of all multiwebs. This generalizes Kasteleyn's theorem and the result of [Douglas, Kenyon, Shi: Dimers, webs, and local systems, Trans. AMS 2023]. We study connections with positive traces, and define the associated probability measure on multiwebs. By careful choice of connection we can thus encode the "free fermionic" subvarieties for vertex models such as the $6$-vertex model and $20$-vertex models, and in particular give determinantal solutions. We also find for each multiweb system an equivalent scalar system, that is, a planar bipartite graph $H$ and a local measure-preserving mapping from dimer covers of $H$ to multiwebs on $G$. We identify a family of positive connections as those whose scalar versions have positive face weights.
We discuss how to construct limit shapes for the domino tiling model (square lattice dimer model) and $5$-vertex model, in appropriate polygonal domains. Our methods are based on the harmonic extension method of [R. Kenyon and I. Prause, Gradient variational problems in $\mathbb{R}^2$, Duke Math J. 2022].
In 2015, Vladimir Fock proved that the spectral transform, associating to an element of a dimer cluster integrable system its spectral data, is birational by constructing an inverse map using theta functions on Jacobians of spectral curves. We provide an alternate construction of the inverse map that involves only rational functions in the spectral data.
For a planar bipartite graph $\mathcal G$ equipped with a $\mathrm{SL}_n$-local system, we show that the determinant of the associated Kasteleyn matrix counts "$n$-multiwebs" (generalizations of $n$-webs) in $\mathcal G$, weighted by their web-traces. We use this fact to study random $n$-multiwebs in graphs on some simple surfaces.
We prove a new integrability principle for gradient variational problems in $\mathbb{R}^2$, showing that solutions are explicitly parameterized by $κ$-harmonic functions, that is, functions which are harmonic for the laplacian with varying conductivity $κ$, where $κ$ is the square root of the Hessian determinant of the surface tension.
We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile ``orientations'' determine the tiling. We associate to a tiling an underlying bipartite planar graph $G$ and its corresponding Kasteleyn matrix $K$. If $G$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences.
Given a graph $G$ and collection of subgraphs $T$ (called tiles), we consider covering $G$ with copies of tiles in $T$ so that each vertex $v\in G$ is covered with a predetermined multiplicity. The multinomial tiling model is a natural probability measure on such configurations (it is the uniform measure on standard tilings of the corresponding "blow-up" of $G$). In the limit of large multiplicities we compute asymptotic growth rate of the number of multinomial tilings. We show that the individual tile densities tend to a Gaussian field with respect to an associated discrete Laplacian. We also find an exact discrete Coulomb gas limit when we vary the multiplicities. For tilings of ${\mathbb Z}^d$ with translates of a single tile and a small density of defects, we study a crystallization phenomena when the defect density tends to zero, and give examples of naturally occurring quasicrystals in this framework.