Piet Lammers (Paris), Rob Van den Berg (Amsterdam), Joseph Chen (Paris), Barbara Dembin (Strasbourg), Trishen Gunaratnam (Mumbai and Bengaluru), Léonie Papon (Wien), Eveliina Peltola (Aalto and Bonn), Rémi Rhodes (Marseille).

Piet Lammers (Paris): GFF convergence of the six-vertex model for $-1 \leq \Delta \leq -1/2$.

The six-vertex model is a paradigmatic example of an integrable planar model, particularly after Lieb's resolution of its anti-ferroelectric and ferroelectric phases in 1967 using the Bethe Ansatz. Over the past fifty years, deeper analyses of the model have revealed profound insights into the structure of two-dimensional integrable systems, most notably through the development of the Yang-Baxter equation, quantum groups, and transfer matrices. In a recent joint project with H. Duminil-Copin, K. Kozlowski, and I. Manolescu, we proved convergence of the six-vertex model to $\sigma(\Delta) \times \Gamma$, where $\Gamma$ is the normalised full-plane Gaussian free field, and where $\sigma(\Delta)^2 = 2 / \arccos \Delta$. The result may also lead to applications in related models, such as the critical planar random-cluster model with $1\leq q \leq 4$ and the Ashkin-Teller model.

Rob Van den Berg (Amsterdam): Forest fire models and percolation.

In this talk I will discuss some versions, or rather variations, of the classical Drossel-Schwabl model of forest fires on a two-dimensional, possibly infinite, grid (lattice). The sites (vertices) of the grid are possible locations of trees. In each of the versions that we discuss, vacant sites become occupied at rate 1. In some versions, each site can be ignited (`hit by lightning') at a tiny rate, in which case the entire occupied cluster of that vertex is immediately destroyed (i.e. becomes vacant). In some other versions, only sites on the boundary of a large finite grid are ignited. In still other versions, an occupied cluster is burnt as soon as it contains at least $N$ vertices, where $N$ is a, typically huge, parameter value. Finally, in some versions new trees can grow on burnt sites, while in other versions this is not the case.
In papers with Pierre Nolin (and one with Demeter Kiss and Pierre Nolin) we showed for some of these models that, if we start with all sites vacant, `spontaneous emergence of fire lanes' takes place, which `substantially limit the spread of fires'. I plan to make the above statements more precise, give an idea of the proofs (which make heavily use of results in near-critical percolation), and speculate a bit about possible extensions, improvements and generalisations.

Joseph Chen (Paris) : Entropic repulsion in low temperature lattice models.

Entropic repulsion refers to when an ordinarily localized surface becomes delocalized in the presence of a floor. A classical setting to study this is the Ising model with Dobrushin boundary conditions: consider a box in $\mathbb{Z}^3$ with plus boundary conditions on the bottom side, and minus on the other five sides of the box. Then there is a surface separating the plus boundary from the minus boundary. Fröhlich and Spencer proved in '87 that the height of this surface diverges; we prove it diverges logarithmically with the size of the box.
Studying the shape of the surface is harder. For this we turn to a class of height functions which includes the Discrete Gaussian. Looking from the bird's eye view, we show that the global shape of the surface scales to either a Wulff shape or the whole box (depending on the specific integer size of the box), and appropriately rescaling the surface along the flat sides of this shape yields a Ferrari-Spohn diffusion.

Barbara Dembin (Strasbourg): Minimal surfaces in random environments.

We consider surfaces of $\mathbb{Z}^d$ in $\mathbb{R}$ and a random environment $\eta$ defined on $\mathbb{Z}^d \times \mathbb{R}$. We are interested in surfaces $\varphi$ that minimize the sum of their elastic energy (the $\ell_2$-norm over $\mathbb{Z}^d$ of the gradient of the surface) and the noise on the surface $\sum_v \eta_{v, \varphi_v}$. When the noise is a fractional Brownian field, we obtain the values of the exponents related to energy and spatial fluctuations.
Joint work with Dor Elboim and Ron Peled.

Trishen Gunaratnam (Mumbai and Bengaluru): The Markov property for $\varphi^4_3$ on cylinders.

In the '60s and '70s, Nelson proved that the Markov property for Euclidean random fields, such as the Gaussian Free Field, is sufficient to reconstruct quantum fields on Minkowski space. Despite overwhelming success in 2d to analyse non-gaussian fields, this approach is notoriously difficult to carry out in 3d. Softer methods exist, but they often give an implicit description of fundamental objects -- such as the Hamiltonian of the theory.
I will talk about joint work with Nikolay Barashkov where we give the first proof of the Markov property for one the simplest 3d non-gaussian models -- the $\varphi^4_3$ model on cylinders. Along the way, we establish a stronger property that is a toy version of Segal's axioms, allowing us to glue different $\varphi^4_3$ models by integrating along an appropriate boundary measure. As an application, we prove novel fundamental spectral properties of the $\varphi^4_3$ Hamiltonian.

Léonie Papon (Wien): On the scaling limit of interfaces in the critical planar Ising model perturbed by a magnetic field.

In this talk, I will present some recent results on the scaling limit of interfaces in the critical planar Ising model perturbed by a magnetic field. I will first consider the case when the field is a deterministic function. In this case, I will show that in the so-called near-critical regime and when the Ising model has Dobrushin boundary conditions, the interface separating $+1$ and $-1$ spins has a scaling limit whose law is conformally covariant and absolutely continuous with respect to SLE$_3$. Its limiting law is a massive version of SLE$_3$ in the sense of Makarov and Smirnov. I will also discuss the scaling limit of this interface when the magnetic perturbation is not near-critical.
In the second part of the talk, I will discuss some ongoing work with Fenglin Huang and Aoteng Xia in which we look at the case when the magnetic field is given by a collection of iid centered Gaussian random variables, one for each vertex. In this setting, in the near-critical regime, we show that almost surely in the disorder, the scaling limit of the quenched law of the $\pm 1$ interface is absolutely continuous with respect to SLE$_3$. We then show that this contrasts with the scaling limit of the quenched law of the collection of nested spin loops, which turns out to be almost surely singular with respect to nested CLE$_3$. This also contrasts with the deterministic case where it is known that in the near-critical regime, any subsequential limit of the collection of nested spin loops is absolutely continuous with respect to nested CLE$_3$.

Eveliina Peltola (Aalto and Bonn): Towards a conformal field theory for critical planar interfaces.

For a number of critical lattice models in 2D statistical physics, it has been proven that scaling limits of interfaces (with suitable boundary conditions) are described by random conformally invariant curves, called Schramm-Loewner evolutions (SLE). So-called partition functions of these SLEs (which also encode macroscopic crossing probabilities) can be regarded as specific correlation functions in the conformal field theory (CFT) associated to the lattice model in question. Although it is not clear how to define the latter mathematically, one can still make sense of many of the properties predicted for these CFTs. I give an overview of how one can rigorously connect 2D statistical physics and CFT in this way.

Rémi Rhodes (Marseille): Liouville Theory.

The study of second-order phase transitions in two-dimensional statistical physics models led, in the 1980s, to the emergence of conformal field theories (CFT). The development of the conformal bootstrap in physics brought remarkable advances in the understanding and solution of these theories. However, a rigorous mathematical construction of CFTs and a complete implementation of the bootstrap program remain major open challenges.
Over the past twenty years, the probabilistic approach to Liouville conformal field theory (LCFT) has achieved important progress. It has not only provided a mathematical construction of the model and a full bootstrap solution, but also made explicit the various structural components expected of a field theory and their interplay: path integral formulation, Segal functoriality, state–field correspondence, representation of the symmetry algebra on a Hilbert space, and the Operator Product Expansion (OPE).
Liouville theory thus stands as a genuine case study for understanding the mathematical structure of conformal field theories. The aim of this talk is therefore to show how these concepts are organized and interact within the framework of Liouville theory.

Back to the main page