## Organizers: Thierry Bodineau, Piet Lammers, Yilin Wang

**Friday May 5**

**14:00 - 16:00**

(Amphi Motchane)

**Paul Dario**(CNRS)

**Localization and delocalization for a class of degenerate convex grad phi interface model.**

In this talk, we will consider a classical model of random interfaces known as the grad phi (or Ginzburg-Landau) model. The model first received rigorous consideration in the work of Brascamp-Lieb-Lebowitz in 1975. Since then, it has been extensively studied by the mathematical community and various aspects of the model have been investigated regarding for instance the localization and delocalization of the interface, the hydrodynamical limit, the scaling limit, large deviations etc. Most of these results were originally established under the assumption that the potential encoding the definition of the model is uniformly convex, and it has been an active line of research to extend these results beyond the assumption of uniform convexity. In this talk, we will introduce the model, some of its main properties, and discuss a result of localization and delocalization for a class of convex (but not uniformly convex) potentials.

**Trishen Gunaratnam**(University of Geneva)

****

**Dipping our toes in Potts lattice gauge theory**

Potts gauge theories were introduced in the '80s by Kogut, Pearson, Shigemitsu, and Sinclair. They are toy examples of lattice gauge theories that exhibit a confinement-deconfinement phase transition of Wilson loop observables. In 3d, they arise as dual models to 3d Ising and Potts models. In 4d, they exhibit self-duality. In this talk, I will give a gentle introduction to these models and their stochastic geometric representations. I will also discuss how modern percolation theory techniques seems to be useful in analysing these models in 4d (at least, making some modest progress). This is based on ongoing work with Sky Cao.

**Friday Apr. 28**

**14:00 - 16:00**

(Amphi Motchane)

**Catherine Wolfram**(MIT)

**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. In this talk, 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. I will also describe some results and problems that illustrate some of the ways that three dimensions is qualitatively different from two.

****

Jinwoo Sung(University of Chicago)

Jinwoo Sung

**The Minkowski content measure for the Liouville quantum gravity metric**

A Liouville quantum gravity (LQG) surface is a random two-dimensional "Riemannian manifold" that is conjectured to be the scaling limit of a wide variety of random planar graph models. LQG was formulated initially as a random measure space and, more recently, as a random metric space. In this talk, I will explain how the LQG measure can be recovered as the Minkowski content measure for the LQG metric, thereby providing a direct connection between the two formulations for the first time. Our primary tool is the mating-of-trees theory of Duplantier, Miller, and Sheffield, which says that an LQG surface explored by an independent space-filling Schramm–Loewner evolution (SLE) curve is an infinitely divisible metric measure space when. This is joint work with Ewain Gwynne (University of Chicago).

**Friday Apr. 21**

**14:00 - 16:00**

(Amphi Motchane)

**Anton Zorich**(University Paris Cité)

**Random square-tiled surfaces of large genus and random multicurves on surfaces of large genus.**

(joint work with V. Delecroix, E. Goujard and P. Zograf)

I will remind how Maxim Kontsevich and Paul Norbury have counted metric ribbon graphs and how Maryam Mirzakhani has counted simple closed geodesic multicurves on hyperbolic surfaces. Both counts use Witten-Kontsevich correlators (they will be defined in the lecture with no appeals to quantum gravity).

I will present a formula for the asymptotic count of square-tiled surfaces of any fixed genus g tiled with at most N squares as N tends to infinity. This count allows, in particular, to compute Masur-Veech volumes of the moduli spaces of quadratic differentials. A deep large genus asymptotic analysis of this formula, performed by Amol Aggarwal, and the uniform large genus asymptotics of intersection numbers of Witten-Kontsevich correlators, proved by Aggarwal, combined with the results of Kontsevich, Norbury and Mirzakhani, allowed us to describe the structure of a random multi-geodesic on a hyperbolic surface of large genus and of a random square-tiled surface of large genus.

As an application I will count oriented meanders on surfaces of any genus and an asymptotic probability to get a meander by a random identification of endpoints of a random braid on a two-component surface of any genus.

****

**Monday Apr. 17 (Exceptional session)**

**15:00 - 16:00**

(Amphi Motchane)

**Pierre-François Rodriguez**(Imperial College London)

**Scaling in percolation models with long-range correlations**

The talk will present recent progress towards a rigorous understanding of the critical regime associated to continuous phase transitions in the presence of long-range correlations, in dimensions larger than 2. Our findings deal with a 3-dimensional percolation model built from the harmonic crystal, which benefits from a rich interplay with potential theory for the associated diffusion. The results rigorously exhibit the scaling behavior of various observables of interest and unveil the values of the associated critical exponents, which are consistent with scaling theory below the upper-critical dimension (expectedly equal to 6). This confirms various predictions by physicists based on non-rigorous renormalization group techniques, notably that of Weinrib-Halperin concerning the value of the associated correlation length exponent. It also yields a proof of Fisher’s scaling relation for this model. Based on joint works with Alex Drewitz and Alexis Prevost.

**Friday Apr. 14**

**14:00 - 16:00**

(Amphi Motchane)

**Gérard Ben Arous**(NYU)

**High-dimensional limit theorems for Stochastic Gradient Descent: effective dynamics and critical scaling**

This is a joint work with Reza Gheissari (Northwestern) and Aukosh Jagannath (Waterloo), Outstanding paper award at NeurIPS 2022.

We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic (ODE) and diffusive (SDE) limits, with the limit depending dramatically on the former choices. Interestingly, we find a critical scaling regime for the step-size below which the effective ballistic dynamics matches gradient flow for the population loss, but at which, a new correction term appears which changes the phase diagram. About the fixed points of this effective dynamics, the corresponding diffusive limits can be quite complex and even degenerate. We demonstrate our approach on popular examples including estimation for spiked matrix and tensor models and classification via two-layer networks for binary and XOR-type Gaussian mixture models. These examples exhibit surprising phenomena including multimodal timescales to convergence as well as convergence to sub-optimal solutions with probability bounded away from zero from random (e.g., Gaussian) initializations.

**Friday Mar. 24**

**14:00 - 16:00**

(Amphi Motchane)

**Romain Panis**(University of Geneva)

**Translation invariant Gibbs measures and continuity for φ^4_d via random tangled currents**

In this talk I will present recent results obtained in joint work with Trishen Gunaratnam, Christoforos Panagiotis and Franco Severo concerning the study of Gibbs measures of the lattice φ^4_d model on Z^d. We prove that the set of translation invariant Gibbs measures for the φ^4_d model on Z^d has at most two extremal measures at all temperature. We also give a sufficient condition to ensure that the set of all Gibbs measures is a singleton. As an application, we show that the spontaneous magnetisation of the nearest-neighbour φ^4_d model on Z^d vanishes at criticality for d>=3. The analogous results were established for the Ising model in the seminal works of Aizenman, Duminil-Copin, and Sidoravicius (Comm. Math. Phys., 2015), and Raoufi (Ann. Prob., 2020) using the so-called

*random current representation*introduced by Aizenman (Comm. Math. Phys., 1982). Our proof relies on a new corresponding stochastic geometric representation for the φ^4_d model called the

*random tangled current representation*.

**Trishen Gunaratnam**(University of Geneva)

Blume-Capel and the tricritical point （canceled and postponed）

The Blume-Capel model is a ferromagnetic spin system which can be thought of as an Ising model on a percolation cluster (with annealed disorder). In the physics literature, it is well-studied due to its surprisingly rich phase diagram: along a line of critical points, there is a tricritical point that marks the boundary between discontinuous and continuous phase transition. The tricritical point in 2 dimensions is particularly interesting, as it is expected to be in a different universality class, connected with the so-called dilute Ising CFT. I'll describe some recent work with D. Krachun and C. Panagiotis where we rigorously establish the existence of at least one tricritical point for Blume in all dimensions. I'll also describe some thoughts towards the tricritical point in 2D.

**Friday Mar. 10**

**14:00 - 16:00**

(Amphi Motchane)

**Amaury Freslon**(Paris-Saclay)

A glimpse of random walks on (quantum?!) groups

Random walks on groups are nice examples of Markov chains which arise quite naturally in many situations. Their key feature is that one can use the algebraic properties of the group to gain a fine understanding of the asymptotic behaviour. For instance, it has been observed that some random walks exhibit a very sharp phase transition called the cut-off phenomenon. I will first explain this phenomenon on a concrete example, introducing all the necessary material. Then, I will give some ideas on how the setting may be enlarged to encompass more general stochastic processes (called non-commutative Markov chains) by using a strange algebraic structure called a quantum group.

**Angeliki Menegaki**(IHES)

Spectral gap for long-range interactions in harmonic chain of oscillators

We consider one-dimensional chains and multi-dimensional networks of harmonic oscillators coupled to two Langevin heat reservoirs at different temperatures. Each particle interacts with its nearest neighbours by harmonic potentials and all individual particles are confined by harmonic potentials, too. In previous works we investigated the sharp N-particle dependence of the spectral gap of the associated generator in different physical scenarios and for different spatial dimensions. In this talk I will present new results on the behaviour of the spectral gap when considering longer-range interactions in the same model. In particular, depending on the strength of the longer-range interaction, there are different regimes appearing where the gap drastically changes behaviour but even the hypoellipticity of the operator breaks down. This is a joint work with Simon Becker (ETH).

**Thursday Feb. 23**

**11: 00 - 13: 00**

(Centre de conférences Marilyn et James Simons)

**Scott Armstrong**(NYU) Lecture (3/3)

Quantitative homogenization for probabilists

The goal of this informal lecture series is to explain the analysis/PDE point of view of some problems which arise in statistical physics and probability. I will try to argue that the language of elliptic/parabolic homogenization brings a new perspective to a wide range of problems, and that the quantitative "coarse-graining" methods are surprisingly useful and adaptable. We will try to cover the first part of the recent monograph co-written with Tuomo Kuusi (available here: https://arxiv.org/abs/2210.06488), and then proceed based on the interests of the audience.

**Thursday Feb. 16**

**11: 00 - 13: 00**

(Centre de conférences Marilyn et James Simons)

**Scott Armstrong**(NYU) Lecture (2/3)

Quantitative homogenization for probabilists

The goal of this informal lecture series is to explain the analysis/PDE point of view of some problems which arise in statistical physics and probability. I will try to argue that the language of elliptic/parabolic homogenization brings a new perspective to a wide range of problems, and that the quantitative "coarse-graining" methods are surprisingly useful and adaptable. We will try to cover the first part of the recent monograph co-written with Tuomo Kuusi (available here: https://arxiv.org/abs/2210.06488), and then proceed based on the interests of the audience.

**Thursday, Feb. 9**

**11: 00 - 13: 00**

(Centre de conférences Marilyn et James Simons)

**Barbara Dembin**(ETHZ)

Upper tail large deviations for chemical distance in supercritical percolation

We consider supercritical bond percolation on Z^d and study the chemical distance, i.e., the graph distance on the infinite cluster. It is well-known that there exists a deterministic constant μ(x) such that the chemical distance D(0,nx) between two connected points 0 and nx grows like nμ(x). We prove the existence of the rate function for the upper tail large deviation event {D(0,nx)>nμ(x)(1+ϵ),0< - >nx} for d>=3. Joint work with Shuta Nakajima.

**Pieter Lammers**(IHES)

A mass identity for the 2D XY model

The 2D XY model has attracted attention of physicists and mathematicians for several decades. One way to understand this model is through its dual height function. Recent developments make it possible to show that the phase transitions of the two models coincide. At the core of the proof is a new perspective on the Symanzik/Brydges–Fröhlich–Spencer random walk. The talk is based on arXiv:2301.06905 (Bijecting the BKT transition) and arXiv:2211.14365 (A dichotomy theory for height functions).

**Thursday, Feb. 2**(Centre de conférences Marilyn et James Simons)

**Scott Armstrong**(NYU) Lecture (1/3)

Quantitative homogenization for probabilists

The goal of this informal lecture series is to explain the analysis/PDE point of view of some problems which arise in statistical physics and probability. I will try to argue that the language of elliptic/parabolic homogenization brings a new perspective to a wide range of problems, and that the quantitative "coarse-graining" methods are surprisingly useful and adaptable. We will try to cover the first part of the recent monograph co-written with Tuomo Kuusi (available here: https://arxiv.org/abs/2210.06488), and then proceed based on the interests of the audience.

****

Thursday, Jan. 19, 2023(Centre de conférences Marilyn et James Simons)

Thursday, Jan. 19, 2023

**Dmitry Chelkak**(U. Michigan)

Convergence of double-dimers to CLE(4) via isomonodromic tau-functions

The main goal of this talk is to discuss a series of works (Kenyon’11, Dubédat’14, Basok-Ch.’18, Bai-Wan’21) on the convergence of double-dimer loop ensembles in Temperleyan domains to the nested CLE(4). Contrary to the convergence results available for several other lattice models in 2D (LERW/UST, critical Ising and percolation), this approach does not rely upon martingale observables for single interfaces and uses a probabilistic interpretation of a certain SL(2)-isomonodromic tau-function instead.

The plan is to start with a crash introduction on what is known/predicted for the scaling limits of loop O(N) models in 2D – even though this is not directly related to the main subject of the talk – so as to keep a bigger picture in mind and to have more room for informal questions/discussions.

**Jan. 5, 2023 Amphi Motchane 11-13**

Junchen Rong(IHES)

Junchen Rong

Hand-waving introduction to two-dimensional conformal field theory.

I will try to explain the representation theory of the Virasoro algebra and its application to various statistical physics models such as the Ising model and the free compact boson (Gaussian free field) theory. If time permits, I will also discuss the space of c=1 conformal field theories.

**Jiaming Xia**(IHES)

We consider the random field Ising model on Z^2 with external field i.i.d. N(0,\epsilon). I will present that under nonnegative temperatures, the effect of boundary conditions at distance N away on the magnetization in a finite box decays exponentially. I will first talk about the perturbative analysis, which is a crucial tool used in the proof, and then about the similarities in the proofs of the zero temperature case and the positive temperature case. This talk is based on the joint work with Jian Ding.