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

Phase transitions for the 2D two-component Coulomb gas

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

Morris Ang

Proof of the Delfino-Viti conjecture for planar percolation

For critical percolation on the 2D triangular lattice, consider the probability that three points lie in the same cluster. The Delfino-Viti conjecture predicts that in the fine mesh limit, under suitable normalization, this probability converges to the imaginary DOZZ formula from conformal field theory. We prove the Delfino-Viti conjecture, and more generally, obtain the cluster connectivity three-point function of the conformal loop ensemble. Our arguments depend on the coupling between Liouville quantum gravity and the conformal loop ensemble.

Based on joint work with Gefei Cai, Xin Sun, and Baojun Wu.

Ahmed Bou-Rabee

Anomalous diffusion for Brownian motion with random drift

A Brownian particle subject to a random, divergence-free drift will have enhanced diffusion. The correlation structure of the drift determines the strength of the diffusion and there is a critical threshold, bordering the diffusive and superdiffusive regimes. Physicists have long expected logarithmic-type superdiffusivity at thre threshold, and algebraic superdiffusion above it. I will discuss recent [arXiv:2404.01115] and ongoing work with Scott Armstrong and Tuomo Kuusi in which we address these problems using techniques from the theory of stochastic homogenization.

Lucas D’Alimonte

Ornstein—Zernike theory for the near-critical planar random cluster model

In this talk, we will discuss the classical Ornstein-Zernike theory for the random-cluster model (also known as FK percolation). In its modern form, it is a very robust theory, which most celebrated output is the computation of the asymptotically polynomial corrections to the pure exponential decay of the two-points correlation function of the random-cluster model in the subcritical regime. We will present a recent work that extends this theory to the near-critical regime of the two-dimensional random-cluster model, thus providing a precise understanding of the Ornstein-Zernike asymptotics when p approaches the critical parameter pc. The output of this work is a formula encompassing both the critical behaviour of the system when looked at a scale negligible with respect to its correlation length, and its subcritical behaviour when looked at a scale way larger than its correlation length. Based on a joint work with Ioan Manolescu.

Diederik van Engelenburg

Arm exponents of the high dimensional Ising model

Based on joint work with Christophe Garban, Romain Panis and Franco Severo.

In this talk, I will focus on behavior of the Ising model in high dimensions (d ≥ 4).

Widom proposed that thermodynamic quantities follow power laws governed by critical exponents, and above the upper critical dimension d_c = 4, these exponents reduce to the mean-field values (matching those on trees or complete graphs). I will talk about a recent work about the so-called one-arm event (the origin connects to distance n) in the FK-Ising model,

We observe that this exponent depends on the boundary condition: for wired boundary conditions, we prove that this probability decays up to constants as n^(-1) for d ≥ 4 (by the Edwards-Sokal coupling, this probability is the magnetization at the origin), whereas in infinite volume we prove that it decays as n^(-2) for d ≥ 6. You may be wondering at this point why there is a discrepancy in the dimension. To avoid spoiling the potential surprise, I will provide the answer only during the talk.

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

The limiting law of the Discrete Gaussian level-lines

We will present a recent result with Joe Chen on the low temperature (2+1)D integer-valued Discrete Gaussian (ZGFF) model. The level lines were conjectured to have cube-root fluctuations near the sides of the box, mirroring the Solid-On-Solid picture. The new results resolve this and further recover the joint limit law of the top level-lines near the sides of the box, which is a product of Ferrari—Spohn diffusions.

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

Kieran Ryan

Ground states in the XXZ chain and the Lorentz mirror model with loop weight 2

We present some work in progress on infinite volume ground states of the 1-D quantum Heisenberg XXZ model. The work uses a representation of the model as a probabilistic continuous time model of loops due to Ueltschi. The Lorentz mirror model with loop weight 2 can be thought of as a discrete time version of this loop model, and has a simple coupling with the 6-vertex model.

We use a recent result of Glazman and Lammers on the six-vertex model and a further, new coupling to show that for the Lorentz loop model, certain finite volume measures converge to a common infinite volume measure, which exhibits no infinitely long loops, and has slow decay of connection probabilities.

We then prove an analogous theorem to Glazman and Lammers for a “space-time” version of the six-vertex model. Using this, we prove the same statement in Ueltschi’s loop model as for the mirror model, and its consequences for the XXZ model: certain finite volume, finite temperature states converge to a common infinite volume ground state, with slowly decaying spin-spin correlations. In particular, as with Glazman and Lammers’ work, these arguments do not use any integrability or exact solutions.

Lukas Schoug

A first passage metric of the Gaussian free field

Using the Gaussian free field (GFF) and its level sets, we construct a conformally invariant random metric on a planar domain, in which, heuristically, the length of a curve is given by the maximal value of the GFF on said curve. Aside from the interesting geometry of the metric, one motivation for its study is in the context of string theory: in the 1980s, Polyakov introduced Liouville quantum gravity (LQG) — a one-parameter family of random surfaces — with the goal of describing a conformally invariant string theory, however, LQG is conformally covariant instead. We conjecture that our constructed metric should arise as a renormalised limit of LQG and describe the model that Polyakov aimed to construct.

Franco Severo

On the supercritical phase of the φ^4 model

The φ^4 model is a real-valued spin system with quartic
potential. This model has deep connections with the classical Ising
model, and both are expected to belong to the same universality class.
We construct a random cluster representation for φ^4, analogous to that
of the Ising model. For this percolation model, we prove that local
uniqueness of macroscopic cluster holds throughout the supercritical
phase. The corresponding result for the Ising model was proved by
Bodineau (2005) and serves as the crucial ingredient in renormalization
arguments used to study fine properties of the supercritical behaviour,
such as surface order large deviations, the Wulff construction and
exponential decay of truncated correlations. The unboundedness of spins
in the φ^4 model imposes considerable difficulties when compared with
the Ising model. This is particularly the case when handling boundary
conditions, which we do by relying on the recently constructed random
current representation of the model. Joint work with Trishen Gunaratnam,
Christoforos Panagiotis and Romain Panis.

Aoteng Xia

Phase transition and Near-critical behavior of random field Ising model

In this talk, we discuss the random-field Ising model where the external field is given by i.i.d. random variables with mean 0 and variance \eps^2. The Imry–Ma prediction from 1975 suggests that this model exhibits a phase transition in spontaneous magnetization in dimensions three and higher, but not in two dimensions. We present recent efforts and key ideas toward a rigorous proof of this prediction. In addition, we explore interesting near-critical phenomena in two dimensions when \eps is allowed to decay with the system size $N$.
This talk is based on joint work with Jian Ding, Chenxu Hao, Fenglin Huang, Yu Liu, and Léonie Papon.