Two-dimensional lattice models/Modèles sur réseaux bidimensionnels

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- Patrik Ferrari,
*Dimers and orthogonal polynomials: connections with random matrices*- Gaussian Unitary Ensemble of random matrices (GUE): eigenvalue distribution, determinantal correlations and Hermite polynomials, interlacing structure in GUE minors.
- Totally Asymmetric Simple Exclusion Process (TASEP): interlacing structure in continuous time TASEP, determinantal correlations and Charlier polynomials (evtl. discrete time and Krawtchouk polynomials).
- 2+1 dimensional anisotropic growth: particle dynamics, interface interpretation, dynamics on random tiling, two projections: TASEP and random tiling (evtl. diffusion scaling limit and GUE minors, discrete time and Aztec diamond).

- Nikolai Reshetikhin,
*TBA* - Béatrice de Tilière,
*Dimers and random surfaces: combinatorics and local statistics*The dimer model represents the adsorption of di-atomic molecules on the surface of a crystal. It is one of the rare non-trivial, exactly solvable of statistical mechanics. In this mini-course, we shall first prove Kasteleyn's founding result, giving an exact formula for the partition function. We shall then explain the very fruitful connection, due to Thurston, between dimers on bipartite graphs and random surfaces. Our final goal is to give an overview of the paper of Kenyon, Okounkov and Sheffield, giving a full understanding of the model on infinite periodic, bipartite graphs: free energy, Gibbs measures, phases of the model.

- Sylvie Corteel,
*Domino tilings, lattice paths and plane overpartitions* - Julien Dubédat,
*Fluctuations of dimer height functions*We review some results describing the scaling limit of the height function in a dimer model in terms of the Gaussian free field and related objects.

- Eric Nordenstam,
*The shuffling algorithm and tilings of an Aztec diamond*Recently there has been much activity around dynamics on interlaced particles, for example Borodin & Ferrari, Warren, Dieker & Warren. One of these models suggests itself naturally in the context of domino tilings of a certain shape in the plane called the Aztec Diamond.

- Paul Zinn-Justin,
*Exactly solvable models of tilings and Littlewood–Richardson coefficients*There are various known combinatorial rules for computing Littlewood-Richardson coefficients. A particularly attractive one is the so-called puzzles of Knutson and Tao. Puzzles are related to a model of random tilings, the so-called square-triangle tiling model. We discuss the consequences of the quantum integrability of the latter. If time allows, we shall introduce a more general model, of square-triangle-rhombus tilings, which allows for “equivariant” generalizations of Littlewood-Richardson coefficients.

- Arvind Ayyer,
*Descending Plane Partitions and Permutations*The connections between alternating sign matrices and descending plane partitions had been pondered upon ever since it was discovered that they were counted by the same numbers. A more refined conjecture proposed by Mills, Robbins and Rumsey in 1983 states that the number of ASMs with k -1's is the same as that of descending plane partitions with k special parts. As a first step towards this understanding, we exhibit a natural bijection between descending plane partitions with no special part and permutations.

- Mihai Ciucu,
*The field of average tile orientations in random tilings with holes*The study of random tilings of planar lattice regions goes back to the solution of the dimer model in the 1960's by Kasteleyn, Temperley and Fisher, but received new impetus in the early 1990's, and has since branched out in several directions in the work of Cohn, Kenyon, Okounkov, Sheffield, and others.

In this talk, we focus on the interaction of holes in random tilings, a subject inspired by Fisher and Stephenson's 1963 conjecture on the rotational invariance of the monomer-monomer correlation on the square lattice. In earlier work, we showed that the correlation of a finite number of holes on the triangular lattice is given asymptotically by a superposition principle closely paralleling the superposition principle for electrostatic energy.

We now take this analogy one step further, by showing that the discrete field determined by considering at each unit triangle the average orientation of the lozenge covering it converges, in the scaling limit, to the electrostatic field.

Our proof involves a variety of ingredients, including combinatorial arguments, Laplace's method for the asymptotics of integrals, Newton's divided difference operator, and hypergeometric function identities.

- Filippo Colomo,
*Arctic curves for the domain-wall six vertex model*In the first part of the seminar the derivation of the Arctic Curve of the domain-wall six vertex model will be reviewed. The second part will be devoted to a short discussion of:

- application to the limit shape of q-enumerated large Alternating Sign Matrices; in particular, as q→0, the limit shape tends to a nontrivial limiting curve;
- possibility of extending the result to the anti-ferroelectric regime;
- the model with a “free boundary”.

- Bertrand Eynard,
*A matrix model for plane partitions and tilings*We show how to design a matrix model for counting plane partitions or dimers, with rather arbitrary weights and boundary conditions. Once we have a matrix model, it becomes clear that limit statistical properties are necessarily matrix models limit laws, and also, before taking any limit, random matrices techniques (orthogonal polynomials, determinantal formulae, integrability, loop equations) provide a huge toolbox which one may use for plane partitions and dimers. In particular, it not only allows to find limit shapes, but also all orders corrections can be computed.

- Paul Fendley,
*Classical and Quantum Self-duality*A wealth of valuable information follows from knowing that the Boltzmann weights of a classical lattice model satisfy Kramers-Wannier duality. For example, seemingly different types of ordering can be seen to be equivalent, and often critical points can be located exactly. In this talk I explain how such duality has a topological origin, arising from the connection of integrable lattice models with conformal field theory and link invariants such as the Jones polynomial. This enables duality to be extended to a huge class of classical lattice height models. It also greatly simplifies the construction of quantum “net” models, by demanding that the ground state wavefunction satisfy an analogous quantum self-duality.

- Clément Hongler and Stanislav Smirnov,
*Energy density correlations in the 2D Ising model*The 2D Ising model at criticality is considered a classical example of conformal invariance in statistical mechanics, which is used in deriving many of its properties. However, up till recently no mathematical proof has ever been given, and even physics arguments support (a priori weaker) Möbius invariance.

We consider Ising model in a simply connected planar domain with + or free boundary conditions, and show that in the scaling limit the energy density field is conformally covariant. We suggest a way to compute its correlations, giving a rigorous derivation of the Conformal Field Theory predictions and exhibiting a nice connection with hyperbolic geometry. Moreover, we give improved formulas for the discrete energy density, which (to the best of our knowledge) are not attainable by CFT techniques.

The proofs rely on discrete holomorphic fermions.

- Rinat Kedem,
*Statistical mechanical models and positivity conjectures in cluster algebras*I will give an introduction to of the use of fusion relations coming from generalized Heisenberg models in proving positivity conjectures in cluster algebras. Fusion relations are regarded as discrete evolution equations, and they are integrable in the Liouville sense. They can also be viewed as an algeberaic object known as cluster algebras. Integrability provides a tool for giving an exact solution. Conjectures about the positivity of solutions follows from the explicit solution.

- Michael Kozdron,
*Using multiple SLE to explain a certain observable in the 2d Ising model*The Schramm-Loewner evolution (SLE) is a one-parameter family of random growth processes that has been successfully used to analyze a number of models from two-dimensional statistical mechanics. Currently there is interest in trying to formalize our understanding of conformal field theory using SLE. Smirnov recently showed that the scaling limit of interfaces of the 2d critical Ising model can be described by SLE(3). The primary goal of this talk is to explain how a certain non-local observable of the 2d critical Ising model studied by Arguin and Saint-Aubin can be rigorously described using multiple SLE(3) and Smirnov's result. As an extension of this result, we explain how to compute the probability that a Brownian excursion and an SLE(κ) curve, with κ between 0 and 4, do not intersect.

- Pierre Mathieu,
*Combinatorics of RSOS paths*Every minimal model in conformal field theory can be viewed as the scaling limit of a restricted-solid-on-solid (RSOS) model at criticality. States in irreducible modules of the minimal model

*M(p',p)*can be described combinatorially by paths that represent configurations in the corresponding RSOS model, dubbed RSOS*(p',p)*. These paths are in one-to-one correspondence with tableaux with prescribed hook differences. For*p'=2*, these are tableaux with successive ranks in a prescribed interval, which are known to be related to the Bressoud paths (whose generating function is the sum side of the Andrews-Gordon identity). We show how the RSOS*(2,p)*paths can be directly related to these paths. Generalizing this construction, we arrive at a representation of RSOS paths in terms of generalized Bressoud paths (for*p>2p'*). These new paths have a simple weighting and a natural particle interpretation. This then entails a natural particle spectrum for RSOS paths, which can be interpreted in terms of the kinks and breathers of the restricted sine-Gordon model. - Philippe Nadeau,
*Fully Packed Loop configurations in a triangle*We are interested in enumerating Fully Packed Loop (FPL) configurations on the square grid with a given link pattern. We will prove a general decomposition formula expressing such a configuration of size n in terms of configurations of size n-1. The coefficients appearing in this decomposition are defined in terms of certain FPL configurations in a triangle. We will enumerate such triangle configurations for various boundary conditions. In one case, FPL configurations in the triangle turn out to be counted by the famous Littlewood-Richardson coefficients.

- Soichi Okada,
*Two-parameter deformation of multivariate hook product formulae*Gansner uses the Hillman-Grassl correspondence to prove the hook product formulae for multivariate (i.e., trace) generating functions of ordinary or shifted reverse plane partitions of a given shape. In this talk, we give another proof and a two-parameter deformation of Gansner's formulae, following an approach of Okounkov and Reshetikhin. Also we present a conjectural deformation of Peterson-Proctor's hook product formula for P-partitions on d-complete posets.

- Xavier Viennot,
*The cellular Ansatz and the configurations B.A.BA*We propose a unified geometrico-algebraic setting for the study, and in particular the enumeration, of various 2D lattice structures appearing in statistical mechanics, such as alternating sign matrices (ASM), fully packed loops (FPL), rhombus tilings, plane partitions, non-intersecting paths or certain tableaux interpreting the stationary probabilities of the PASEP model.

The cellular Ansatz is in two steps. The first one consists in the “geometrization” of the commutation relations defining a general quadratic algebra with some rewriting rules on planar cells of a square lattice. For a certain quadratic algebra with 4 generators and 8 parameters, this planarization leads to the configurations B.A.BA which contain virtually all the configurations above. Some new bijections seem to appear.

In a second step, the cellular Ansatz consists in the representation of the generators of a quadratic algebra with some operators acting on a class of combinatorial objects. This representation leads “automatically” to the construction of bijections. The classical Robinson-Schensted-Knuth (RSK) correspondence is an example (in its “local” form defined by Fomin with the so-called growth diagrams). The bijection between permutations and the alternative tableaux related to the PASEP is a second example (in its “local” form defined with “Laguerre histories”). The extension to B.A.BA configurations is open.

Finally, these configurations B.A.BA seem to be good candidates for a combinatorial interpretation of the integers appearing in the exact computation of the correlation function of the XXZ spins-1/2 Heisenberg chains by Kitanine, Maillet, Slavnov, Terras.

- Lauren Williams,
*Tableaux combinatorics for the asymmetric exclusion process and Askey-Wilson polynomials*We introduce some new tableaux, and connect them to both the asymmetric exclusion process (ASEP) and Askey-Wilson polynomials. The ASEP is a model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites with open boundaries. In its most general form, particles may enter and exit at the left with probabilities alpha and gamma, and they may exit and enter at the right with probabilities beta and delta. In the bulk, the probability of hopping left is q times the probability of hopping right. Our first result is a formula for the stationary distribution of the ASEP with all five parameters general, in terms of tableaux. Our second result is a related “combinatorial” formula for the moments of Askey-Wilson polynomials. Since the 1980's there has been a great deal of work giving combinatorial formulas for moments of classical orthogonal polynomials. However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials. This is joint work with Sylvie Corteel.