Hendrik Weber

Scaling limits of interacting particle systems via SPDE methods

2nd February 2026
The derivation of hydrodynamic and fluctuation limits for interacting particle systems is a classical theme in probability theory and mathematical physics.

In this talk, I will describe a new approach to such scaling limits inspired by modern techniques for singular stochastic partial differential equations, in particular the theory of regularity structures. As a concrete case study, I will discuss the weakly asymmetric simple exclusion process on a circle and its convergence, under diffusive scaling, to the Kardar–Parisi–Zhang (KPZ) equation. This convergence is a classical result, first established by Bertini and Giacomin in the 1990s.

Traditionally, the proof relies on the non-linear Gärtner (Cole–Hopf) transform, which linearises the problem at the discrete level. I will explain how one can instead analyse the particle system directly, without relying on such integrable structures, by adapting SPDE methods to a discrete, jump-driven setting. This leads to a flexible and robust framework that is expected to extend to a broad class of interacting particle systems.

Karsten Matthies

Timescales for the invariance principle of the random Lorentz gas

9th February 2026
We study the timescale T(r) of the invariance principle for a Lorentz gas with particle size r and improve results by obtained by Lutsko and Toth (2020). The work includes improved geometric recollision estimates and a discussion of the coupling of stochastic processes introduced in the original result. Joint work with Raphael Winter.

Irene Ayuso Ventura

Imry–Ma phenomenon for the hard-core model on ℤ²

16th February 2026
I will present recent joint work with Leandro Chiarini, Tyler Helmuth, and Ellen Powell on the hard-core model on ℤ², a model of independent sets on the square lattice. We show that under weak random disorder, this model has no phase transition in two dimensions. This behavior is known as the Imry–Ma phenomenon, whose most classical example is the random-field Ising model. Our proof is inspired by the Aizenman–Wehr argument for the random-field Ising model, but relies on spatial symmetries rather than internal spin symmetries.

Kirstin Strokorb

Graphical models for infinite measures with applications to extremes and Lévy processes

23rd February 2026
Conditional independence and graphical models are well studied for probability distributions on product spaces. We propose a new notion of conditional independence for any measure on the punctured Euclidean space that explodes at the origin. The importance of such measures stems from their connection to infinitely divisible and max-infinitely divisible distributions, where they appear as Lévy measures and exponent measures, respectively. We characterize independence and conditional independence for such measures in various ways through kernels and factorization of a modified density, including a Hammersley-Clifford type theorem for undirected graphical models. As opposed to the classical conditional independence, our notion is intimately connected to the support of the measure. Structural max-linear models turn out to form a Bayesian network with respect to our new form of conditional independence. Our general theory unifies and extends recent approaches to graphical modeling in the fields of extreme value analysis and Lévy processes.

This talk is based on joint work: S. Engelke, J. Ivanovs, and K. Strokorb. Graphical models for infinite measures with applications to extremes. Ann. Appl. Probab. 35(5): 3490-3542 (October 2025). DOI: 10.1214/25-AAP2201

Joost Jorritsma

The critical percolation window in uniformly grown random graphs

2nd March 2026
We study graphs in which vertices 1,...,n arrive sequentially, and in which vertex j connects independently to earlier vertices with probability β/j. Bollobás, Janson, and Riordan (‘05) proved that the phase transition in β beyond which a giant component appears is of infinite order, in sharp contrast to the classical mean-field transition in Erdős–Rényi random graphs.

In this talk we determine the asymptotic order of the largest component at the critical value β_c. Moreover, for sequences β_n → β_c, we describe the critical window and uncover several surprising features, including a secondary phase transition and bounded susceptibility (average component size) throughout the window.

The proofs rely on a coupling between a component exploration process and a branching random walk with two killing barriers.

Joint work with Pascal Maillard and Peter Mörters.

Julian Hofstadler

(Some) theoretical results for Markov chain- and Power Iteration Monte Carlo algorithms

9th March 2026
In this edition of PIMS I will talk about two of my ongoing projects in Bath regarding the theory of Monte Carlo algorithms. In the first part of the talk, the focus is on Markov chain-based algorithms, and how Poisson’s equation can be used to analyse these. In the second part of the talk, Power Iteration methods in the context of neutron transport are studied.

Emmanuel Kammerer

The height of the infection tree

16th March 2026
Consider an SIR model on the complete graph starting with one infected vertex and n sane vertices. We draw an edge between two vertices when one infects another. What does the tree look like at the end of the epidemic? This kind of tree fits into the framework of uniform attachment trees with freezing, a model of random trees which generalises uniform attachment trees where, besides the uniform attachment mechanism, we introduce a "freezing" mechanism where new vertices cannot attach to frozen vertices. We obtain the scaling limit of the total height of the infection tree depending on the infection rate. The asymptotic behaviour of the total height satisfies a phase transition of order 2. This talk is based on a joint work with Igor Kortchemski and Delphin Sénizergues.

Jonathon Peterson

Limit Theorems for self-interacting random walks: a Ray-Knight approach

20th March 2026 (Friday!), 14:15
Self-interacting random walks are discrete models for random motion where the transition probabilities of the walk depend in some way on the past of the walk. Such random walks are often very difficult to study because of their non-Markovian nature. However, for certain families of self-interacting random walks it has been observed that there is a Markovian structure embedded in the directed edge local times (the counts of how many times the walk has crossed each edge in a particular direction). This Markovian structure can often be analyzed to prove scaling limits for the directed edge local times of the walk; such results are called generalized Ray-Knight theorems. The challenge, however, is then to use these generalized Ray-Knight theorems to deduce scaling limits for the random walk. In this talk I will discuss some of my results in this direction, focusing on recent results with Kosygina, Mountford, and Mareche proving functional scaling limits for recurrent self-interacting random walks via such Ray-Knight methods.

John Fernley

The Second Phase Transition of the Contact Process on a Random Regular Graph

23rd March 2026
The regular tree corresponds to the random regular graph as its local limit. For this reason the famous double phase transition of the contact process on regular tree has been seen to correspond to a phase transition on the large random regular graph, at least at the first critical value. We find a phase transition on that large finite graph at the second critical value: between linear reinfections and reinfections following a long healthy period.