Recent advances in loop models and height functions

Juhan Aru Extremal distance and conformal radius of a CLE_4 loop.

Conformal loop ensembles (CLE_\kappa) are random collections of loops in a simply-connected planar domain. For 8/3 < \kappa \leq 4 they are characterised by conformal invariance + a certain Markovian property, and are conjectured to appear as scaling limits of different loop O(n) models at critical parameters. We consider CLE_4 in the unit disk and take a closer look at the loop L surrounding the origin. We will discuss how to calculate the law of the "distance" of L to the origin, the law of the "distance" of L to the boundary of the unit disk, and the joint law of both of these distances. This is joint work with Titus Lupu and Avelio Sepulveda.

Alexander Glazman Six-vertex and Ashkin-Teller models: order/disorder phase transition.

The 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)

Subhajit Goswami Anomalous diffusion on the GFF landscape.

Diffusion in a random potential with log-correlations is a model of great interest studied in different contexts in statistical mechanics, condensed matter physics and population biology among others. In two-dimension a canonical example of a log-correlated field is the Gaussian free field (GFF). In this talk I will present a family of random walk models on the square lattice indexed by an inverse temperature parameter where the the underlying transition probabilities are governed by a sample of the two-dimensional GFF with appropriate boundary conditions. The random walk jumps to a neighbour v with probability proportional to the exponential of the product of inverse temperature and field value at v. As such this is a model of random walk in random environment (RWRE) where the underlying environment is strongly correlated. It has been predicted in the physics literature that this walk is sub-diffusive and furthermore the diffusive exponent exhibits a dynamic phase transition around a certain critical temperature. I will discuss some rigorous results where we have been able to partially confirm these predictions including rigorous derivations of precise diffusive exponents at all temperatures.
Based on a joint work with Marek Biskup and Jian Ding.

Marcin Lis Spins, percolation and height functions.

To capture apparent similarities in combinatorial representations of several two-dimensional models of statistical mechanics, we propose a four-parameter family of models (two integer and two real parameters) which specializes to these known cases after a proper tuning. The models come in the form of a spin system, a bond percolation model and a height function coupled together in a natural way analogous to the Edwards-Sokal coupling for the random cluster and Potts model. We discuss basic properties of these models and their mutual relationships.
As special cases we recover the standard Potts and random cluster model, the six-vertex model and loop O(n) model, the random current, double random current and XOR Ising model.
Moreover, based on a non-rigorous computation of Nienhuis of the variance of the height function in the six-vertex model, we predict that the scaling limit of contours in the symmetric six-vertex model is a specific level set of the Gaussian free field. This is a generalization of Wilson's conjecture about the scaling limit of the interfaces in the XOR Ising model.

Ellen Powell A characterisation of the 2d Gaussian Free Field.

I will discuss a joint work with Nathanael Berestycki and Gourab Ray, in which we prove that a random distribution in two dimensions satisfying conformal invariance and a natural domain Markov property must be a multiple of the Gaussian free field. This result holds subject only to a mild moment assumption.

Ron Peled The loop O(n) model - the main conjectures and some recent progress.

The loop O(n) model is a model for non-intersecting, self-avoiding loops. Its version on the hexagonal lattice is especially intriguing due to its conjectured critical behavior, on which many detailed predictions are available in the physics literature, and due to the fact that it generalizes several other models of interest. I will introduce the hexagonal lattice model, explain the main conjectures regarding it and the limited rigorous progress made. I will then focus on a recent work which uses the "XOR trick" to show the existence of macroscopic loops in a certain subset of the phase diagram. This is the first rigorous proof that macroscopic loops emerge for a positive area of the parameter space.
Joint work with Nicholas Crawford, Alexander Glazman and Matan Harel.

Ron Peled Site percolation on planar graphs and circle packings.

Color each site of a planar graph blue with probability p and red with probability 1-p, independently among sites. For which values of p is there an infinite connected component of blue sites? This is a classical problem in statistical physics and probability theory which is well studied when the planar graph is a lattice. Recently, Itai Benjamini made several conjectures on the problem for general planar graphs, relating it to the behavior of the simple random walk there. Understanding the problem for general planar graphs is also suggested by the study of the loop O(n) model. I will discuss a connection of the problem with circle packings and the progress that it yields.

Andrew Elvey Price Counting planar maps equipped with a height function.

We define a height function on a graph to be a labelling of the vertices such that adjacent labels differ by exactly 1. We study planar quadrangulations equipped with a height function, which we call height-labelled quadrangulations. EP and Guttmann showed that these are in bijection with 4-valent maps equipped with an Eulerian orientation, which were counted exactly by Kostov in 1999 as a specialisation of what he called the six vertex model on a random lattice. I will describe an alternative method with which we count height-labelled quadrangulations directly. This leads us to a much simpler form of the solution than what was found by Kostov. We also generalise our method to determine the exact distribution of labels amongst all height-labelled quadrangulations.
This is joint work with Mireille Bousquet-Melou.

Lorenzo Taggi Non-decay of correlations in the dimer model and phase transition in lattice permutations in Z^d, d>2, via reflection positivity.

In this talk we consider the dimer model and the model of random lattice permutations. Our first result is that the two-point correlation function of the dimer model in Z^d does not decay with the distance when d > 2. This confirms physicists conjectures and shows that in dimensions greater than two the dimer model behaves drastically differently than in two dimensions, in which case it is known that correlations decay polynomially. Our second result involves a self-avoiding walk interacting via hard-core repulsion with a system of mutually-disjoint (possibly oriented and coloured) self-avoiding loops and double dimers, a model which is related to the quantum Bose gas through a representation due to Feynman 1953 and to the loop O(N) model. We prove the occurrence of a phase transition in this model when d > 2, namely the existence of a regime in which the self-avoiding walk is 'long' and the distance between its end-points grows linearly with the diameter of the box. These results follow from the derivation of an 'Infrared-ultraviolet' bound from a new probabilistic settings, with walks entering into the system from some 'virtual' vertices and interacting with coloured loops, and from the property of reflection positivity of such a gas of loops and walks.

Martin Tassy Delocalization of uniform graph homomorphisms from \(Z^2\) to \(Z\).

Graph homomorphisms from the \(Z^2\) lattice to \(Z\) are functions on \(Z^2\) whose gradients equal \(1\) in absolute value. Following arguments presented by Scott Sheffield in Random surface which are adapted and simplified to the present settings, we will show that this model delocalizes in two dimensions, having no translation-invariant Gibbs measures for the uniform sampling subject to boundary conditions. We also obtain additional results in higher dimensions including the facts that every ergodic Gibbs measure is extremal. Finally we will discuss what part of the arguments can be generalized to other height function models such as the loop O(n) model.