Mini-courses

Ellen Powell - Characterising the Gaussian free field

I will talk about recent techniques and results concerning axiomatic characterisations of the Gaussian free field in arbitrary dimensions.

Greta Panova - Asymptotic Algebraic Combinatorics slides1 slides2

Algebraic Combinatorics studies objects and quantities originating in Representation Theory and Algebra via combinatorial tools. It has seen many connections with Probability and Statistical Mechanics both in terms of the objects (e.g. plane partitions are dimer covers of a region in the hexagonal grid) and tools (symmetric functions, R matrices, cluster algebras etc). I will cover the necessary background (Young tableaux, Schur functions etc) and some of the corresponding techniques to study dimers. I will then introduce some of the outstanding problems in Algebraic Combinatorics concerning (asymptotic) analysis of structure constants, show some approaches via Probability and Statistical Mechanics and present further open problems which may inspire application and development of other such tools.

Ioan Manolescu - Exploring the critical phase of 2D FK-percolation

In these lectures we will study FK-percolation (also called the random-cluster model) on the square lattice Z². Our focus will be on recent results describing the phase transition, namely:

• its sharpness;
• its continuity/discontinuity depending on the cluster parameter q;
• the large-scale rotational invariance of the critical models.

These results have been obtained through a wide variety of techniques; of particular interest will be the relationship between FK-percolation and the six-vertex model, and the treatment of the latter using the transfer matrix formalism.

Talks

Nathanaël Berestycki - Near-critical dimers and massive SLE slides

A programme initiated by Makarov and Smirnov is to describe near-critical scaling limits of planar statistical mechanics models in terms of massive SLE and/or Gaussian free field. We consider here the dimer model on the square or hexagonal lattice with doubly periodic weights, which is known to have non Gaussian limits in the whole plane.

In joint work with Levi Haunschmid (TU Vienna) we obtain the following results: (a) we establish a rigourous connection with the massive SLE₂ constructed by Makarov and Smirnov; (b) we show that the convergence of the height function in arbitrary bounded domains subject to Temperleyan boundary conditions, and that the scaling limit is universal; and (c) we prove conformal covariance of the scaling limit. Our techniques rely on Temperley’s bijection and the “imaginary geometry” approach developed in earlier work with Benoit Laslier and Gourab Ray, as well as a new exact discrete Girsanov identity on the triangular lattice.

Time-permitting we will discuss conjectures relating this model to the sine-Gordon model.

Tomas Berggren - Geometry of the doubly periodic Aztec dimer model

We will discuss the doubly periodic Aztec diamond dimer model of growing size, with arbitrary periodicity and only mild conditions on the edge weights. In this limit we see three types of macroscopic regions – known as rough, smooth and frozen regions. We will discuss how the geometry of the arctic curves, the boundary of the rough region, can be described in terms of an associated amoeba and an action function. In particular, we determine the number of frozen and smooth regions and the number of cusps on the arctic curves. We will also discuss the convergence of local fluctuations to the appropriate translation-invariant Gibbs measures. Joint work with Alexei Borodin.

Sunil Chhita - The Two-Periodic Aztec diamond slides

Domino tilings of the two-periodic Aztec diamond exhibit interesting statistical mechanical behaviors – a limit shape emerges separating three macroscopic interfaces, known as frozen, rough and smooth that depend on the local statistics. We survey some of the main results on this model, as well as recent progress on understanding the rough-smooth boundary. This is based on joint work with Duncan Dauvergne and Thomas Finn.

Filippo Colomo - Arctic curves of the four vertex model slides

We consider the four-vertex model with a particular choice of fixed boundary conditions. In the scaling limit, the model exhibits the limit shape phenomenon, with the emergence of an arctic curve separating a central disordered region from six frozen “corners” of ferroelectric or anti ferroelectric type. We determine the analytic expression of the arctic curve. The derivation is based on the exact evaluation of suitable correlation functions, discriminating spatial transition from order to disorder, in terms of the partition function of some discrete log-gas associated to Hahn polynomials. As a by-product, we also deduce that the arctic curve’s fluctuations are governed by the Tracy-Widom distribution.

Joint work with I.N. Burenev, A. Maroncelli, and A.G. Pronko

Christophe Garban - Surface law and charge rigidity for the Coulomb gas on ℤ^d

I will start by introducing and motivating the (two-component) Coulomb gas on the d-dimensional lattice ℤ^d. I will then present some puzzling properties of the fluctuations of this Coulomb gas. The connection of this model with integer-valued fields and compact-valued spin systems will be emphasised through the talk. This is based on joint works with Avelio Sepúlveda and David Dereudre.

Terrence George - Move-reduced graphs on a torus slides

We determine which bipartite graphs embedded in a torus are move-reduced. In addition,we classify equivalence classes of such move-reduced graphs under square/spider moves.This extends the class of minimal graphs on a torus studied by Goncharov–Kenyon, and gives a toric analog of Postnikov’s results on a disk. Joint work with Pavel Galashin.

Sasha Glazman - Delocalisation of height functions slides

We will discuss the delocalisation of height functions in two dimensions. The main example will be the six-vertex model when a,b ≤c ≤ a+b. Our approach applies also to integer-valued Lipschitz functions (loop O(2) model): we show delocalisation when ½ ≤ x² ≤ 1. The main idea behind the argument is a novel FKG property for a joint distribution of spin and percolation configurations. It enables the use of the non-coexistence theorem of Zhang and Sheffield. If time permits, we will also discuss how to derive from this a new proof for the continuity of the phase transition in the random-cluster model.

In the delocalised regime, the height functions are expected to converge to the Gaussian free field. This is so far known only at the line a² + b² = c² (dimer model) and in its neighbourhood.

Based on a joint work with Piet Lammers.

Vadim Gorin - Boundary limits for the six-vertex model slides

Take a random configuration of (a,b,c)-weighted six-vertex model in a very large planar domain. What does it look like near a straight segment of the boundary? We investigate this question on the example of the model in N*N square with Domain Wall Boundary Conditions and find that the answer depends on the value of Δ=(a²+b²-c²)/(2ab): there is a single universal limiting object for all Δ<1 and a richer class of limits at Δ>1.

David Keating - Double dimers, coupled tilings, and LLT polynomials slides

We study k-tuples of tilings of the Aztec diamond in which we assign a weight to each tuple depending on the number of ‘interactions’ between the tilings. There are several ways to view these tilings: as a double dimer model with an interaction between the dimers, as an LLT process generalizing the usual Schur process, or as a certain integrable vertex model. We will define the coupled tilings relating them to each of these settings. Then we will describe a generalization of the domino shuffling algorithm to the coupled tilings.

Richard Kenyon - Dimers and 3-webs

Associated to a GL_n-local system on a graph on a surface is a combinatorial formula enumerating traces of n-webs in the graph, in terms of a determinant of an associated matrix, the Kasteleyn matrix. We explain this formula and show how, for SL_3, can be used to enumerate webs of given topological types, in the case of graphs in simple surfaces. This is based on joint work with Dan Douglas and Haolin Shi.

Kalle Kytölä - Local fields of lattice models and conformal field theories slides

A fundamental feature any conformal field theory (CFT) is a representation of Virasoro algebra on the space of local fields of the theory: Virasoro algebra describes the effect of (infinitesimal) conformal transformations on the fields. By combining discrete holomorphicity and Sugawara-type constructions, we showed that also the spaces of suitably defined lattice local fields of the Ising model and the discrete Gaussian free field (DGFF) have Virasoro representations. Similarly, a Virasoro representation for local fields of the double-dimer model has been recently found by Adame-Carrillo. Having a Virasoro representation on the local fields of these lattice models opens up the possibility of a direct correspondence, preserving the Virasoro actions, between lattice model local fields and CFT local fields. We will discuss such a 1-to-1 correspondence for local fields of the DGFF and a Fock space of local fields of a free boson CFT (which is generated under operator product expansions by the gradient of the continuum Gaussian Free Field). The correspondence exactly describes the scaling limit: correlation functions of the local fields of the DGFF converge to correlation functions of the corresponding fields in the CFT, when renormalized by lattice mesh to the power of their L₀+ \bar{L}₀ eigenvalues.

based on joint work with Hongler & Viklund + with Adame-Carrillo & Behzad

Marcin Lis - Conformal invariance of critical double random currents slides

The double random current (DRC) model is a natural percolation model whose geometric properties are intimately related to spin correlations of the Ising model. In two dimensions, it moreover carries an integer valued height function on the graph, called the nesting field. We study the critical DRC model on bounded domains of the square lattice. We fully describe the joint scaling limit of the (primal and dual) DRC clusters and the nesting field as the lattice mesh size vanishes. We prove that the nesting field becomes the Dirichlet Gaussian free field (GFF) in this limit, and that the outer boundaries of the DRC clusters with free boundary conditions are the conformal loop ensemble with κ=4 (CLE4) coupled to that GFF. Moreover, we also show that the inner boundaries of the DRC clusters form a two-valued local set with values ∓ 2λ,(2√2 ∓ 2)λ for the field restricted to a CLE4 loop with boundary value ±2λ. Our proof is a combination of exact solvability of the Ising model, new crossing estimates for the DRC model (which does not posses the FKG property), and a careful analysis of the structure of two-valued local sets of the continuum GFF.

This is joint work with Hugo Duminil-Copin and Wei Qian.

Alessandra Occelli - Universality of multi-component stochastic systems slides

Universality classes are identified by exponents and scaling functions that characterise the macroscopic behaviour of the fluctuations of the thermodynamical quantities of interest in a microscopic system. When considering multi-component systems different universality classes might appear according to the asymmetry of the interactions. To see which universality classes might appear, we outline the approach of Nonlinear Fluctuation Hydrodynamics Theory (NLFHT), introduced by Spohn 2014. As an example, we study the equilibrium fluctuations of an exclusion process evolving on the discrete ring with three species of particles named A,B and C. We prove that proper choices of density fluctuation fields (that match of those from nonlinear fluctuating hydrodynamics theory) associated to the conserved quantities converge, in the large N limit, to a system of stochastic partial differential equations, that can either be the Ornstein-Uhlenbeck equation or the Stochastic Burgers’ equation.

Mirjana Vuletic - Free boundary Schur process slides

Free boundary Schur process is a generalization of the original Schur process of Okounkov and Reshetikhin, with no restrictions on boundary partitions. This model after “shift-mixing” is a Pfaffian process. This talk will be about models and asymptotic results related to this process. The limiting behavior for these models is given by some new deformations of universal distributions of Schur processes. The talk is based on a joint work with D. Betea, J. Bouttier, and P. Nejjar.

Catherine Wolfram - Large deviations for the 3D dimer model

A dimer tiling of Z^d is a collection of edges such that every vertex is covered exactly once. In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. I will explain how to formulate the large deviations principle in 3D, show simulations, and explain some of the ways that we use a smaller set of tools (e.g. Hall’s matching theorem or a double dimer swapping operation) in our arguments. Time permitting, I will also explain some stories that illustrate qualitative differences between two and three dimensions.