Piet Lammers

Towards GFF convergence for the six-vertex model

29th September 2025
The six-vertex model is a height function model that serves as a unifying framework for several two-dimensional statistical mechanics systems. In this talk, I will present a proof of a long-standing conjecture asserting that the height function converges, in a certain parameter regime, to the Gaussian Free Field (GFF). The proof combines techniques from different areas of mathematics: at its core is a soft analysis of the transfer matrix, which notably avoids reliance on the Bethe Ansatz. This analysis is made rigorous through probabilistic tools, including the Fortuin-Kasteleyn-Ginibre (FKG) inequality and Russo-Seymour-Welsh (RSW) theory. This is joint work with Hugo Duminil-Copin, Karol Kozlowski, and Ioan Manolescu. I will also discuss a closely related conjecture—that the Fortuin-Kasteleyn percolation associated with the six-vertex model converges in the scaling limit to a Conformal Loop Ensemble, CLE(κ).

Tom Klose

Asymptotic Exit Problems for a Singular Stochastic Reaction-Diffusion Equation

6th October 2025
We consider a singular stochastic reaction-diffusion equation with a cubic non-linearity on the 3D torus and study its behaviour as it exits a domain of attraction of an asymptotically stable point. Mirroring the results of Freidlin and Wentzell in the finite-dimensional case, we relate the logarithmic asymptotics of its mean exit time and exit place to the minima of the corresponding (quasi-)potential on the boundary of the domain. The challenge, in our setting, is that the stochastic equation is singular such that its solution only lives in a Hölder–Besov space of distributions. The proof accordingly combines a classical strategy with novel controllability statements as well as continuity and locally uniform large deviation results obtained via the theory of regularity structures. This is joint work with Ioannis Gasteratos (TU Berlin).

Blaine Van Rensburg

Life in flux: models of population dynamics in changing environments

13th October 2025
This talk is split into two parts. In the first part, I will present a framework for analysing eco-evolutionary PDE models. We apply this framework to a non-local and non-linear PDE model of a prey-population structured by a trait representing their risk-tolerance and determine the asymptotic behavior of the moments of the trait-distribution in the limit of small segregrational variance. Next, I will discuss ongoing work on a two-type branching process model of plants with seedbanks (subject to occasional disasters that kill all plants but not seeds). By studying an embedded branching processes in a random environment, we find a condition that determines if the process has a positive probability of survival or almost surely goes extinct. I will also introduce results related to increasingly critical environments which requires an alternative approach.

John Haslegrave

Two-type annihilation and other reactions on graphs

27th October 2025
Suppose equal numbers of two types of particles move randomly on a finite graph, perhaps at different speeds, with collisions between opposite types resulting in annihilation. How many movements will it take for all particles to be eliminated? This model, introduced by Cristali et al., is surprisingly difficult to analyse, with even the case of the complete graph being previously unknown.

We obtain exact asymptotics for the complete graph and up-to-constant bounds for expander graphs. In the latter case, upper and lower bounds are both non-trivial and use different techniques. I shall discuss how the methods apply to a general class of dissipative particle system, but the specific setup described above creates additional difficulties. This is joint work with Peter Keevash (Oxford).

Benoit Dagallier

The Polchinski dynamics: an introduction

3rd November 2025
I will introduce the Polchinski dynamics (or flow), a general framework to study asymptotic properties of statistical mechanics and field theory models inspired by renormalisation group ideas. The Polchinski dynamics has appeared recently under different names, such as stochastic localisation, and in very different contexts (Markov chain mixing, optimal transport, functional inequalities...) Here I will motivate its construction from a physics point of view and mention a few applications. In particular, I will explain how the Polchinski flow can be used to generalise Bakry and Emery’s Γ2 calculus to obtain functional inequalities (e.g. Poincaré, log-Sobolev) in physics models which are typically high-dimensional and non-convex. The talk is based on a review paper with Roland Bauerschmidt and Thierry Bodineau.

Trishen Gunaratnam

The supercritical phase of the φ⁴ model

10th November 2025
The φ⁴ model is a generalization of the Ising model to a system with unbounded spins that are confined by a quartic potential. A natural random cluster representation for the model arises by considering the sign field, which is distributed as an Ising model in a random environment. I will talk about a proof of local uniqueness of macroscopic clusters throughout the supercritical phase of this percolation model. This serves as a crucial step towards establishing fine properties of the supercritical phase of the spin model via renormalization arguments. Based on joint work with Christoforos Panagiotis, Romain Panis, and Franco Severo.

Tara Trauthwein

CLTs in Stochastic Geometry: Localization for Sums of Scores

17th November 2025
We consider objects in stochastic geometry which are built on Poisson point processes. These can be spatial random graphs, or more complicated models like interacting particle systems. It is well-established that if these objects exhibit a certain kind of 'local' behaviour, e.g. stabilization, then one can show that asymptotically in their size or number of points, many of their statistics like total edge lengths will approach a Gaussian distribution. In this talk, we will look at sums of score functions, where score functions exhibit what is called 'localization'. This means that each score is close in law to a score which only sees a local neighbourhood of the point set. Under these conditions, which are weaker than previous requirements, we show quantitative CLTs under Berry-Esseen type rates. As an illustration, we apply our results to the Random Sequential Adsorption model, which is an interacting particle system.

Matthew Tointon

Finitary escape probabilities for random walks on vertex-transitive graphs

24th November 2025
Varopoulos famously showed that the simple random walk on a vertex-transitive graph is transient if and only if the number of vertices in a ball of radius r grows faster than quadratically in r. I will describe a quantitative, finitary refinement of this result, joint with Romain Tessera, in which we give sharp upper and lower bounds on the probability that the random walk escapes to distance r without first returning to its starting point, resolving and extending a conjecture of Benjamini and Kozma from 2002. This also yields a "gap" in Varopoulos’s result: there exists a universal c > 0 such that if the random walk is transient then its probability of escaping to infinity is at least c.

Min Liu

Last Passage Percolation with Thick Boundaries

1st December 2025
We consider the exponential last passage percolation (LPP) with thick two-sided boundary that consists of a few inhomogeneous columns and rows. Ben Arous and Corwin previously studied the limit fluctuations in this model except in a critical regime, for which they predicted that the limit distribution exists and depends only on the most dominant columns and rows. In this article, we prove their conjecture and identify the limit distribution explicitly in terms of a Fredholm determinant of a 2 by 2 matrix kernel. This result leads in particular to an explicit variational formula for the one-point marginal of the KPZ fixed point for a new class of initial conditions. Our limit distribution is also a novel generalization of and provides a new numerically efficient representation for the Baik-Rains distribution.

Alexander Glazman

Planar percolation and applications

8th December 2025
We witness many phase transitions in everyday life (e.g. ice melting to water). The mathematical approach to these phenomena revolves around the percolation model: given a graph, call each vertex open with probability p independently of the others and look at the subgraph induced by open vertices. Benjamini and Schramm conjectured in 1996 that, at p=1/2, on any planar graph, either there are no infinite connected components or infinitely many.

We prove a stronger version of this conjecture for all planar graphs. In particular, we show that every unimodular invariantly amenable planar graph has p_c ≥ 1/2. We then use this to establish fractal macroscopic behaviour in the loop O(n) model.