Double-dimer Model Research Articles

Discrete Symplectic Fermions on Double Dimers and Their Virasoro Representation

AbstractA discrete version of the conformal field theory of symplectic fermions is introduced and discussed. Specifically, discrete symplectic fermions are realised as holomorphic observables in the double-dimer model. Using techniques of discrete complex analysis, the space of local fields of discrete symplectic fermions on the square lattice is proven to carry a representation of the Virasoro algebra with central charge $$-2$$ - 2 .

Macroscopic loops in the 3d double-dimer model

Macroscopic loops in the 3d double-dimer model

Combinatorics of the double-dimer model

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to a problem in Donaldson-Thomas and Pandharipande-Thomas theory. The proof of our recurrence requires generalizing work of Kenyon and Wilson; specifically, lifting their assumption that the nodes of the graph are black and odd or white and even.

Site-Monotonicity Properties for Reflection Positive Measures with Applications to Quantum Spin Systems

We consider a general statistical mechanics model on a product of local spaces and prove that, if the corresponding measure is reflection positive, then several site-monotonicity properties for the two-point function hold. As an application, we derive site-monotonicity properties for the spin–spin correlation of the quantum Heisenberg antiferromagnet and XY model, we prove that spin-spin correlations are point-wise uniformly positive on vertices with all odd coordinates—improving previous positivity results which hold for the Cesàro sum. We also derive site-monotonicity properties for the probability that a loop connects two vertices in various random loop models, including the loop representation of the spin O(N) model, the double-dimer model, the loop O(N) model and lattice permutations, thus extending the previous results of Lees and Taggi (2019).

Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)

Building upon recent results of Dubédat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations \Omega^\delta to a simply connected domain \Omega\subset\mathbb C we prove the convergence of probabilities of cylindrical events for the double-dimer loop ensembles on \Omega^\delta as \delta\to 0 . More precisely, let \lambda_1,\dots,\lambda_n\in\Omega and L be a macroscopic lamination on \Omega\setminus\{\lambda_1,\dots,\lambda_n\} , i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities P_L^\delta that one obtains L after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on \Omega^\delta converge to a conformally invariant limit P_L as \delta \to 0 , for each L . Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety Hom (\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C) and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do not use any RSW-type arguments for double-dimers. The limits P_L of the probabilities P_L^\delta are defined as coefficients of the isomonodromic tau-function studied in [7] with respect to the Fock–Goncharov lamination basis on the representation variety. The fact that P_L coincides with the probability of obtaining L from a sample of the nested CLE(4) in \Omega requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.

Dominos in hedgehog domains

We introduce a new class of discrete approximations of planar domains that we call “hedgehog domains”. In particular, this class of approximations contains two-step Aztec diamonds and similar shapes. We show that fluctuations of the height function of a random dimer tiling on hedgehog discretizations of a planar domain converge in the scaling limit to the Gaussian Free Field with Dirichlet boundary conditions. Interestingly enough, in this case the dimer model coupling function satisfies the same Riemann-type boundary conditions as fermionic observables in the Ising model. In addition, using the same factorization of the double-dimer model coupling function as in [18], we show that in the case of approximations by hedgehog domains the expectation of the double-dimer height function is harmonic in the scaling limit.

Site Monotonicity and Uniform Positivity for Interacting Random Walks and the Spin O(N) Model with Arbitrary N

We provide a uniformly-positive point-wise lower bound for the two-point function of the classical spin $O(N)$ model on the torus of $\mathbb{Z}^d$, $d \geq 3$, when $N \in \mathbb{N}_{>0}$ and the inverse temperature $\beta$ is large enough. This is a new result when $N>2$ and extends the classical result of Fr\"ohlich, Simon and Spencer (1976). Our bound follows from a new site-monotonicity property of the two-point function which is of independent interest and holds not only for the spin $O(N)$ model with arbitrary $N \in \mathbb{N}_{>0}$, but for a wide class of systems of interacting random walks and loops, including the loop $O(N)$ model, random lattice permutations, the dimer model, the double dimer model, and the loop representation of the classical spin $O(N)$ model.

The Expectation Value of the Number of Loops and the Left-Passage Probability in the Double-Dimer Model

We study various statistical properties of the double-dimer model, a generalization of the dimer model, on rectangular domains of the square lattice. We take advantage of the Grassmannian representation of the dimer model, first to calculate the probability distribution of the number of nontrivial loops around a cylinder, which is consistent with the previously known result, and then to calculate the expectation value of the number of loops surrounding two faces and the left-passage probability, both in the discrete and the continuum cases. We also briefly explain the calculation of some related observables. As a by-product, we obtain the partition function of the dimer model in the presence of two and four monomers, and a single monomer on the boundary.

The Kashaev Equation and Related Recurrences

The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev equation, which has its roots in the Ising model and in the study of relations among principal minors of a symmetric matrix. Certain solutions of the hexahedron recurrence restrict to solutions of the Kashaev equation. We characterize the solutions of the Kashaev equation that can be obtained by such a restriction. This characterization leads to new results about principal minors of symmetric matrices. We describe and study other recurrences whose behavior is similar to that of the Kashaev equation and hexahedron recurrence. These include equations that appear in the study of s-holomorphicity, as well as other recurrences which, like the hexahedron recurrence, can be related to cluster algebras.

Double dimers, conformal loop ensembles and isomonodromic deformations

The double-dimer model consists in superimposing two independent, identically distributed perfect matchings on a planar graph, which produces an ensemble of non-intersecting loops. In [20], Kenyon established conformal invariance in the small mesh limit by considering topological observables of the model parameterized by SL _2 (\mathbb C) representations of the fundamental group of the punctured domain. The scaling limit is conjectured to be CLE _4 , the Conformal Loop Ensemble at \kappa = 4 [36]. In support of this conjecture, we prove that a large subclass of these topological correlators converge to their putative CLE _4 limit. Both the small mesh limit of the double-dimer correlators and the corresponding CLE _4 correlators are identified in terms of the -functions introduced by Jimbo, Miwa and Ueno [14] in the context of isomonodromic deformations.

The free-fermionic [formula omitted] loop model, double dimers and Kashaev's recurrence

We study a two-color loop model known as the C2(1) loop model. We define a free-fermionic regime for this model, and show that under this assumption it can be transformed into a double dimer model. We then compute its free energy on periodic planar graphs. We also study the star-triangle relation or Yang–Baxter equations of this model, and show that after a proper parametrization they can be summed up into a single relation known as Kashaev's relation. This is enough to identify the solution of Kashaev's relation as the partition function of a C2(1) loop model with some boundary conditions, thus solving an open question of Kenyon and Pemantle [29] about the combinatorics of Kashaev's relation.

Dimers in Piecewise Temperleyan Domains

We study the large-scale behavior of the height function in the dimer model on the square lattice. Richard Kenyon has shown that the fluctuations of the height function on Temperleyan discretizations of a planar domain converge in the scaling limit (as the mesh size tends to zero) to the Gaussian Free Field with Dirichlet boundary conditions. We extend Kenyon’s result to a more general class of discretizations. Moreover, we introduce a new factorization of the coupling function of the double-dimer model into two discrete holomorphic functions, which are similar to discrete fermions defined in Smirnov (Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, 2006; Ann Math (2) 172:1435–1467, 2010). For Temperleyan discretizations with appropriate boundary modifications, the results of Kenyon imply that the expectation of the double-dimer height function converges to a harmonic function in the scaling limit. We use the above factorization to extend this result to the class of all polygonal discretizations, that are not necessarily Temperleyan. Furthermore, we show that, quite surprisingly, the expectation of the double-dimer height function in the Temperleyan case is exactly discrete harmonic (for an appropriate choice of Laplacian) even before taking the scaling limit.

Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts

Let $\mathcal{M}$ be the moduli space of rank 2 stable torsion free sheaves with Chern classes $c_i$ on a smooth 3-fold $X$. When $X$ is toric with torus $T$, we describe the $T$-fixed locus of the moduli space. Connected components of $\mathcal{M}^T$ with constant reflexive hulls are isomorphic to products of $\mathbb{P}^1$. We mainly consider such connected components, which typically arise for any $c_1$, "low values" of $c_2$, and arbitrary $c_3$. In the classical part of the paper, we introduce a new type of combinatorics called double box configurations, which can be used to compute the generating function $\mathsf{Z}(q)$ of topological Euler characteristics of $\mathcal{M}$ (summing over all $c_3$). The combinatorics is solved using the double dimer model in a companion paper. This leads to explicit formulae for $\mathsf{Z}(q)$ involving the MacMahon function. In the virtual part of the paper, we define Donaldson-Thomas type invariants of toric Calabi-Yau 3-folds by virtual localization. The contribution to the invariant of an individual connected component of the $T$-fixed locus is in general not equal to its signed Euler characteristic due to $T$-fixed obstructions. Nevertheless, the generating function of all invariants is given by $\mathsf{Z}(q)$ up to signs.

Dimers, webs, and positroids

We study the dimer model for a planar bipartite graph N embedded in a disk, with boundary vertices on the boundary of the disk. Counting dimer configurations with specified boundary conditions gives a point in the totally non-negative Grassmannian. Considering pairing probabilities for the double-dimer model gives rise to Grassmann analogs of Rhoades and Skandera's Temperley–Lieb immanants. The same problem for the (probably novel) triple-dimer model gives rise to the combinatorics of Kuperberg's webs and Grassmann analogs of Pylyavskyy's web immanants. This draws a connection between the square move of plabic graphs (or urban renewal of planar bipartite graphs), and Kuperberg's square reduction of webs. Our results also suggest that canonical-like bases might be applied to the dimer model. We furthermore show that these functions on the Grassmannian are compatible with restriction to positroid varieties. Namely, our construction gives bases for the degree 2 and degree 3 components of the homogeneous coordinate ring of a positroid variety that are compatible with the cyclic group action.

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 \({\mathcal{G}}\) is a union of two dimer covers of \({\mathcal{G}}\) . 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.

Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

Recently, Kenyon and Wilson introduced a certain matrix M in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix M−1 is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of M−1. In this paper we prove the two conjectures. As a consequence we obtain that the sum of the absolute values of all entries of M−1 is equal to the number of complete matchings. We also find a bijection between Dyck tilings and complete matchings.

Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

Recently, Kenyon and Wilson introduced a certain matrix M in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix $M^-1$ is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of $M^-1$. In this paper we prove the two conjectures. As a consequence we obtain that the sum of the absolute values of all entries of $M^-1$ is equal to the number of complete matchings. We also find a bijection between Dyck tilings and complete matchings. Récemment, Kenyon et Wilson ont introduit une certaine matrice M afin de calculer des probabilités d'appariement dans ce qu'ils appellent le modèle double-dimère. Ils ont montrè que la valeur absolue de chaque entrée de la matrice inverse $M^-1$ est égal au nombre de pavages de Dyck d'une certaine forme gauche. Ils ont conjecturè deux formules sur la somme des valeurs absolues des entrées dans une rangée ou une colonne de $M^-1$. Dans cet article, nous prouvons les deux conjectures. En conséquence on obtient que la somme des valeurs absolues de toutes les entrées de M^-1 est égale au nombre de couplages parfaits. Nous trouvons aussi une bijection entre les pavages de Dyck et les couplages parfaits.

Double-Dimer Pairings and Skew Young Diagrams

We study the number of tilings of skew Young diagrams by ribbon tiles shaped like Dyck paths, in which the tiles are "vertically decreasing". We use these quantities to compute pairing probabilities in the double-dimer model: Given a planar bipartite graph $G$ with special vertices, called nodes, on the outer face, the double-dimer model is formed by the superposition of a uniformly random dimer configuration (perfect matching) of $G$ together with a random dimer configuration of the graph formed from $G$ by deleting the nodes. The double-dimer configuration consists of loops, doubled edges, and chains that start and end at the boundary nodes. We are interested in how the chains connect the nodes. An interesting special case is when the graph is $\varepsilon(\mathbb Z\times\mathbb N)$ and the nodes are at evenly spaced locations on the boundary $\mathbb R$ as the grid spacing $\varepsilon\to0$.

Boundary partitions in trees and dimers

Given a finite planar graph, a grove is a spanning forest in which every component tree contains one or more of a specified set of vertices (called nodes) on the outer face. For the uniform measure on groves, we compute the probabilities of the different possible node connections in a grove. These probabilities only depend on boundary measurements of the graph and not on the actual graph structure; i.e., the probabilities can be expressed as functions of the pairwise electrical resistances between the nodes, or equivalently, as functions of the Dirichlet-to-Neumann operator (or response matrix) on the nodes. These formulae can be likened to generalizations (for spanning forests) of Cardy’s percolation crossing probabilities and generalize Kirchhoff’s formula for the electrical resistance. Remarkably, when appropriately normalized, the connection probabilities are in fact integer-coefficient polynomials in the matrix entries, where the coefficients have a natural algebraic interpretation and can be computed combinatorially. A similar phenomenon holds in the so-called double-dimer model: connection probabilities of boundary nodes are polynomial functions of certain boundary measurements, and, as formal polynomials, they are specializations of the grove polynomials. Upon taking scaling limits, we show that the double-dimer connection probabilities coincide with those of the contour lines in the Gaussian free field with certain natural boundary conditions. These results have a direct application to connection probabilities for multiple-strand SLE 2 \operatorname {SLE}_2 , SLE 8 \operatorname {SLE}_8 , and SLE 4 \operatorname {SLE}_4 .

Combinatorics of Tripartite Boundary Connections for Trees and Dimers

A grove is a spanning forest of a planar graph in which every component tree contains at least one of a special subset of vertices on the outer face called nodes. For the natural probability measure on groves, we compute various connection probabilities for the nodes in a random grove. In particular, for "tripartite" pairings of the nodes, the probability can be computed as a Pfaffian in the entries of the Dirichlet-to-Neumann matrix (discrete Hilbert transform) of the graph. These formulas generalize the determinant formulas given by Curtis, Ingerman, and Morrow, and by Fomin, for parallel pairings. These Pfaffian formulas are used to give exact expressions for reconstruction: reconstructing the conductances of a planar graph from boundary measurements. We prove similar theorems for the double-dimer model on bipartite planar graphs.