Schedule for: 19w5062 - Dimers, Ising Model, and their Interactions

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The Yang-Baxter equation states equality of certain local partition functions of a vertex model. If the terms of the Yang-Baxter equation are nonnegative, we can turn it into a a Markov map, which randomly maps objects from one side of the identity into objects on the other side. This idea brings a number of nice applications to lozenge tilings and interacting particle systems.

Unlike classical antiferromagnets, quantum antiferromagnetic systems exhibit ground state frustration effects even in one dimension. A case in point is a quantum spin chain with the interaction between neighboring S-spins given by the projection on the two-spins singlet state. This 1D quantum system's ground state bears a close analogy to the self dual 2D Fortuin-Kasteleyn random cluster model, at Q=(2S+1)^2. The corresponding stochastic geometric representation has led to the dichotomy (Aiz-Nachtergale): for each S the ground state exhibits either (i) slow decay of spin-spin correlations (as in the Bethe solution of the Heisenberg S=1/2 antiferromagnet) or (ii) dimerization, manifested in translation symmetry breaking. Drawing on the recent analysis of the phase transition of the FK models (by Duminil-Gagnebin-Harel-Manolescu-Tassion, and Ray-Spinka), we show that in the infinite volume limit for any S>1/2 this SU(2S+1) invariant quantum system has a pair of distinct ground states, each exhibiting spatial energy oscillations, and exponential decay of correlations. (Joint work with H. Duminil-Copin and S. Warzel).

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will explain a recently discovered mysterious property in a variety of stochastic systems ranging from the six-vertex model and to the directed polymers, last passage percolation, Kardar-Parisi-Zhang equation, and Airy sheet. Vaguely speaking, the property says that the multi-point joint distributions are unchanged when some (but not necessarily all!) points of observations are shifted. The property leads to explicit computations for the previously inaccessible joint distributions in all these settings.

In this talk I will review the results on the universality of height fluctuations in interacting dimer models, obtained in collaboration with F. Toninelli and V. Mastropietro in a recent series of papers. The class of models of interest are close-packed dimers on the square lattice, in the presence of small but extensive perturbations that make them non-determinantal. Examples include the 6-vertex model close to the free-fermion point and the dimer model with plaquette interaction. By tuning the edge weights, one can impose a non-zero average tilt for the height function, so that the considered models are in general not symmetric under discrete rotations and reflections. It is well known that, in the determinantal case, height fluctuations in the massless (or `liquid') phase scale to a Gaussian log-correlated field and their amplitude is a universal constant, independent of the tilt. Our main result is the following: when the perturbation strength is sufficiently small, log-correlations survive, with amplitude A that, generically, depends non-trivially and non-universally on the perturbation strength and on the tilt. Moreover, the amplitude A satisfies a universal scaling relation (`Haldane' or `Kadanoff' relation), saying that it equals the anomalous exponent of the dimer-dimer correlation. The proof is based on a combined use of fermionic Renormalization Group techniques, lattice Ward Identities for the lattice model and emergent Ward Identities for the infrared fixed point theory.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider uniformly random lozenge tilings of essentially arbitrary domains and show that the local statistics of this model around any point in the liquid region of its limit shape are given by the infinite-volume, translation-invariant, extremal Gibbs measure of the appropriate slope. In this talk, we outline a proof of this result, which proceeds by locally coupling a uniformly random lozenge tiling with a model of Bernoulli random walks conditioned to never intersect. Central to implementing this procedure is to establish a local law for the random tiling, which states that the associated height function is approximately linear on any mesoscopic scale.

The five-vertex model is a generalization of the lozenge tiling model, in which certain adjacent dimers interact. It is solvable by Bethe Ansatz. We give an explicit expression for the free energy and surface tension, and show how to parameterize limit shapes explicitly with analytic functions. We similarly solve the "isoradial" version of the model. This talk is based on joint works with de Gier, Watson and Prause.

To highlight certain similarities in combinatorial representations of several well known two-dimensional models of statistical mechanics, we introduce and study a new family of models which specializes to these cases after a proper tuning of the parameters. To be precise, our model consists of two independent standard Potts models, with possibly different numbers of spins and different coupling constants (the four parameters of the model), defined jointly on a graph embedded in a surface and its dual graph, and conditioned on the event that the primal and dual interfaces between spins of different value do not intersect. We also introduce naturally related height function and bond percolation models, and we discuss their basic properties and mutual relationship. As special cases we recover the standard Potts and random cluster model, the 6-vertex model and loop O(n) model, the random current, double random current and XOR-Ising model.

Ashkin-Teller model is a classical four-component spin model introduced in 1943. It can be viewed as a pair of Ising models tau and tau’ with parameter J that are coupled by assigning parameter U for the interaction of the products tau*tau’ at every two neighbouring vertices. On the self-dual curve sinh 2J = e^{-2U}, the Ashkin-Teller model can be coupled with the six-vertex model with parameters a=b=1, c=coth 2J and is conjectured to be conformally invariant. The latter model has a height-function representation. We show that the height at a given face diverges logarithmically in the size of the domain when c=2 and remains uniformly bounded when c>2. In the latter case we obtain a complete description of translation-invariant Gibbs states and deduce that the Ashkin-Teller model on the self-dual line exhibits the following symmetry-breaking whenever J < U: correlations of spins tau and tau’ decay exponentially fast, while the product tau*tau’ is ferromagnetically ordered. The proof uses the Baxter-Kelland-Wu coupling between the six-vertex and the random-cluster models, as well as the recent results establishing the order of the phase transition in the latter model. However, in the talk, we will focus mostly on other parts of the proof: - description of the height-function Gibbs states via height-function mappings and T-circuits, - coupling between the Ashkin-Teller and the six-vertex models via an FK-Ising-type representation of these two models. (this is joint work with Ron Peled)

We give a short proof of the discontinuity of phase transition for the random-cluster model on the square lattice with parameter q>4. This result was recently shown by Duminil-Copin, Gagnebin, Harel, Manolescu and Tassion via the so-called Bethe ansatz for the six-vertex model. Our proof also exploits the connection to the six-vertex model, but does not rely on the Bethe ansatz. Our argument is soft and only uses very basic properties of the random-cluster model. Joint work with Gourab Ray.

We study asymptotics of perfect matchings on a large class of graphs called the contracting square-hexagon lattice,whichis constructed row by row from either a row of a square grid or a row of a hexagonal lattice. We assign thegraph periodicedge weights with period 1*n, and consider the probability measure of perfect matchings in which theprobability of eachconfiguration is proportional to the product of edge weights. We show that the partition function ofperfect matchings onsuch a graph can be computed explicitly by a Schur function depending on the edge weights. Byanalyzing the asymptoticsof the Schur function, we then prove the Law of Large Numbers (limit shape) and the CentralLimit Theorem (convergenceto the Gaussian free field) for the corresponding height functions. We also show that thedistribution of certain type ofdimers near the turning corner is the same as the eigenvalues of Gaussian Unitary Ensemble,and explicitly study thecurve separating the liquid region and the frozen region for certain boundary conditions.

In this talk I will describe a correspondence between dimer model on planar bipartite graphs and circle pattern embedding of these graphs, that is, embeddings of these graphs so that each face is cyclic. This correspondence is the key for studying Miquel dynamics, a discrete integrable system on circle patterns. Time permitting, I will explain how Tutte embeddings for resistor networks and s-embeddings for Ising models arise as special cases. This is joint work with Richard Kenyon (Yale University), Wai Yeung Lam (University of Luxembourg) and Marianna Russkikh (MIT).

We discuss a concept of `perfect t-embeddings’, or `p-embeddings', of weighted bipartite planar graphs. (T-embeddings also appeared under the name Coulomb gauges in a recent work of Kenyon, Lam, Ramassamy and Russkikh.) We believe that these p-embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. To support this idea, we first develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon's holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model. Further, given a sequence of (abstract) planar graphs G_n and their p-embeddings T_n onto the unit disc D, assume that (i) the faces of T_n satisfy certain technical assumptions in the bulk of D; (ii) the size of the associated origami maps O_n tends to zero as n grows (again, on each compact subset of D). We prove that (i)+(ii) imply the convergence of the fluctuations of the dimer height functions on G_n (provided that these graphs are embedded by T_n), to the GFF on the unit disc D equipped with the standard complex structure. Though this is not fully clear at the moment, we conjecture that the origami maps O_n are always small in absence of frozen regions and gaseous bubbles, so our theorem can be eventually applied to all such cases. Moreover, the same techniques are applicable in the situation when the limit of the origami maps arising from a sequence of p-embeddings is a Lorenz-minimal surface, in this situation one eventually obtains the GFF in the conformal parametrization of this surface. In a related joint work with Sanjay Ramassamy we argue that such a Lorenz-minimal surface indeed arises in the case of classical Aztec diamonds; a general conjecture is that this should `always' be the case due to a link between p-embeddings and a representation of the dimer model in the Plucker quadric. Time permitting, we also indicate how the theory of t-holomorphic functions specifies to the Ising case and discuss related results on conformal invariance of the Ising model as well as a more general perspective.

We discuss a concept of `perfect t-embeddings’, or `p-embeddings', of weighted bipartite planar graphs. (T-embeddings also appeared under the name Coulomb gauges in a recent work of Kenyon, Lam, Ramassamy and Russkikh.) We believe that these p-embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. To support this idea, we first develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon's holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model. Further, given a sequence of (abstract) planar graphs G_n and their p-embeddings T_n onto the unit disc D, assume that (i) the faces of T_n satisfy certain technical assumptions in the bulk of D; (ii) the size of the associated origami maps O_n tends to zero as n grows (again, on each compact subset of D). We prove that (i)+(ii) imply the convergence of the fluctuations of the dimer height functions on G_n (provided that these graphs are embedded by T_n), to the GFF on the unit disc D equipped with the standard complex structure. Though this is not fully clear at the moment, we conjecture that the origami maps O_n are always small in absence of frozen regions and gaseous bubbles, so our theorem can be eventually applied to all such cases. Moreover, the same techniques are applicable in the situation when the limit of the origami maps arising from a sequence of p-embeddings is a Lorenz-minimal surface, in this situation one eventually obtains the GFF in the conformal parametrization of this surface. In a related joint work with Sanjay Ramassamy we argue that such a Lorenz-minimal surface indeed arises in the case of classical Aztec diamonds; a general conjecture is that this should `always' be the case due to a link between p-embeddings and a representation of the dimer model in the Plucker quadric. Time permitting, we also indicate how the theory of t-holomorphic functions specifies to the Ising case and discuss related results on conformal invariance of the Ising model as well as a more general perspective.

We develop a framework to study the dimer model on Temperleyan graphs embedded on a Riemann surface with finitely many holes and handles. We show that the dimer model can be understood in terms of an object which we call Temperleyan forests and show that if the Temperleyan forest has a scaling limit then the fluctuations of the height one-form of the dimer model also converge. Furthermore, if the Riemann surface is either a torus or an annulus, we show that Temperleyan forests reduce to cycle-rooted spanning forests and show convergence of the latter to a conformally invariant, universal scaling limit. As a consequence, the dimer height one-form fluctuations also converge on these surfaces, and the limit is conformally invariant. A key idea here is the geometric description of the scaling limit of a cycle-rooted spanning forest in the universal cover of the surface, achieved using tools coming in particular from the Fuschian theory of hyperbolic Riemann surfaces. Joint work with Benoit Laslier (Paris) and Gourab Ray (Victoria).

I will discuss the local structure of the boundary at the rough-smooth (or liquid-gas) interface in the two-periodic Aztec diamond. At the frozen-rough interface we have a well-defined boundary path which converges to the Airy process. At the rough-smooth smooth boundary the situation is more complicated although here also we expect to have an Airy process. Which geometric structure converges to the Airy process? At this boundary we see both local and long range structures. There should be a last path among these long range structures that converges to the Airy process as can be seen in pictures. We will not prove this but discuss some results in this direction. This is joint work with Vincent Beffara and Sunil Chhita.

We will discuss some properties of the time process for the boundary of the North polar region of the Aztec diamond, based on analogue results obtained in other models in the KPZ universality class.

We consider the dimer model on a bipartite periodic graph with elliptic weights introduced by Fock. The spectral curves of such models are in bijection with the set of all genus 1 Harnack curves. We prove an explicit and local expression for the two-parameter family of ergodic Gibbs measures and for the slope of the measures. This is work in progress with Cédric Boutillier and David Cimasoni.

For dimers and other models of random surfaces, limit shapes appear when boundary conditions force a certain response of the system. The main mathematical tool to study these responses is a variational principle which states that the limiting profile of the system must maximizes the integral of an entropy function often named surface tension. As a consequence, the strict convexity of the surface tension plays a crucial role as it forces the asymptotic profile which maximizes this integral to be unique. In this talk we will show that all models of Lipschitz random surfaces which are stochastically monotonic must have a strictly convex surface tension (joint with Piet Lammers).

I will discuss how dimer models and other statistical physics models are related to Laplacian determinants, both on the discrete level and on the continuum level. In particular, I will recall the geometric meaning of the so-called zeta-regularized determinant of the Laplacian, as it is defined on a compact surface, with or without boundary. Using an appropriate regularization, we find that a Brownian loop soup of intensity c has a partition function described by the (-c/2)th power of the determinant of the Laplacian. In a certain sense, this means that decorating a random surface by a Brownian loop soup of intensity c corresponds to weighting the law of the surface by the (-c/2)th power of the determinant of the Laplacian. I will then introduce a method of regularizing a unit area LQG sphere, and show that weighting the law of this random surface by the (-c'/2)th power of the Laplacian determinant has precisely the effect of changing the matter central charge from c to c'. Taken together with the earlier results, this provides a way of interpreting an LQG surface of matter central charge c as a pure LQG surface decorated by a Brownian loop soup of intensity c. This is based on joint work with Morris Ang, Minjae Park, and Joshua Pfeffer.

The eight-vertex model is an ubiquitous description that generalizes several spin systems, "ice-type" six-vertex models, and others. In a special "free-fermion" regime, it is known since the work of Fan, Lin, Wu in the late 60s that the model can be mapped to non-bipartite dimers. In this talk I will show how to transform these non-bipartite dimers into bipartite ones, which allows for a deep, rigorous study on generic planar graphs. I will mention a few consequences: computation of long-range correlations and critical exponents, critical regimes, and their "exact" application to Z-invariant regimes on isoradial graphs.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.