First week

*This talks are a continuation of the mini-course of Marcin Lis

Second Week

Mini-Courses

Marcin Lis

The Ising magnetization field and the Gaussian free field

Malin Palö

An introduction to lattice gauge theories and equivalent models

Gauge theories are natural higher-dimensional generalizations of classical spin systems such as the Ising model, the Potts model, the XY model, and the Heisenberg model. As such, many classical tools for spin systems can be generalized to a framework that includes both gauge theories and classical spin systems. Using duality, these representations naturally lead to several other related models, each of which elucidates different properties of the original gauge theory. In this course, we will introduce gauge theories with and without external fields, describe several equivalent models, and show how these together explain some basic properties of gauge theories.

Ron Peled

Magnetization in the random-field Ising and XY models

Jeanne Boursier

TBA

Piet Lammer

GFF convergence of the six-vertex model for -1 <= Delta <= -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) * 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 <= q <= 4 and the Ashkin-Teller model.

Tyler Helmuth

The Arboreal Gas and Fermionic Field Theory

Talks

Michael Aizenman

TBA

Diederik van Engelenburg

TBA

Alexander Glazman

Delocalization of Lipschitz function

We consider integer-valued 1-Lipschitz functions on the triangular lattice. Each pair of vertices having different values gets weight x > 0, and the probability is proportional to the product of the weights. We show that the function delocalizes for all  $x \in [1/sqrt{2}, 1]$ under 0 boundary conditions: the variance of the value at the origin tends to infinity for any increasing sequence of domains exhausting the lattice. We discuss two proofs of this result: via a joint FKG property (with Lammers) and via a general percolation result on planar graphs (with Harel and Zelesko). The point $x=1/sqrt{2}$ is expected to be critical, but this remains open: the localization has been proven only when $x \leq 1/sqrt{3} + eps$.

Janne Junnila

On singularity lines of complex projective structures with real holonomy

I will discuss certain curve systems on the punctured Riemann sphere that are induced by complex projective structures with real holonomy. Such curves appear e.g. in semiclassical limits of the Schramm-Loewner evolution. A central question is whether every isotopy class of a curve system can be realized as the singularity lines of some projective structure, and whether it is unique. With Bonk, Rohde and Wang we answered this affirmatively in the special case when the projective structure is induced by a differentiable Jordan curve passing through the punctures and satisfying the following geodesic property: Every arc on the curve is a hyperbolic geodesic in the domain bounded by all the other arcs. These curves turn out to be Loewner energy minimizing in their isotopy classes, and we also proved that the accessory parameters of the associated Schwarzian derivative satisfy an identity analogous to the famous formula conjectured by Polyakov and proven by Takhtajan and Zograf in the Fuchsian case. In addition to explaining these results, I will also showcase a few examples of other such curve systems whose projective structures can be explicitly described thanks to certain symmetries and holonomical constraints.

Huber Lacoin

TBA

Benoit Laslier

Tilted Solid on Solid is liquid, at least when thawed

The Solid on Solid model is a mainstay of the modelisation of 2D interfaces in the physics literature and it has also received extensive attention in mathematics. In particular a major work of Frölich and Spencer showed that it exhibits a roughening transitions where at low temperature, the interface is extremely localized with O(1) fluctuations at the microscopic scale while at high temperature it delocalize with logarithmic variance.

However, almost all the existing literature focuses on the case where the interface is parallel to the main axis of the underlying lattice, as in a crystal facet, but of course this cannot be the case globally everywhere. We will show that, at least for a model with a small added potential, whenever the interface has a tilt the phenomenology changes completely : at low enough temperature prove that the behavior of the interface is rough (like the high temperature usual case) and provide a full scaling limit for the fluctuations. The main approach is a comparison to the zero-temperature case which can be described as lozenge tilings and a renormalization procedure to understand the asymptotic of an interacting tiling model.

Joint work with Eyal Lubetzky.

Thomas Leble

(Looking for a) phase transition in the two-dimensional one-component plasma

The two-dimensional “one-component plasma” (2DOCP), also known as Coulomb or log-gas, is an important model of statistical physics *in the continuum*. In the 1980’s, physicists have conjectured that it undergoes a phase transition at inverse temperature \approx 140. The nature of this transition is yet to be understood mathematicallly.
My goal is to present recent results about “order” and “disorder” within this model, and to ask for help 🙂

Eyal Lubetzki

TBA

Rémy Mahfouf

Traveling trough (random) critical Ising and dimer models 

In this talk, I will present a new differential approach to understanding how the large-scale geometry of (near-)critical Ising and dimer models evolves as their coupling constants are moved continuously. When this evolution is driven by random perturbations, the question connects to the stability of certain stochastic differential equations (SDEs), allowing the construction of (near-)critical random-bond Ising and dimer models whose critical window in a random environment is significantly larger than in the deterministic setting. The talk is based on arXiv:2509.08928 and ongoing work with Benoît Laslier and Mikhail Basok.

Ellen Powell

TBA

Rémi Rhodes

TBA

Lukas Schoug

TBA

Franco Severo

TBA

Aoteng Xia

TBA