50 years in probability - A workshop in honor of Erwin Bolthausen

From Tuesday 9th of June 2026 to Thursday 11th of June 2026.

Kurt - On the multitype Bolthausen-Sznitman-coalescent and its relatives

The Bolthausen-Sznitman coalescent and its connection with Neveu’s continuous state branching process was the starting point of a series of papers by Bertoin and Le Gall on the many relations between multiple merger coalescents (Λ-coalescents) and continuous state branching processes (CSBPs). In this talk, we consider multitype Λ-coalescents that were recently characerized by Johnston, Kyprianou and Rogers. We present several connections with multitype CSBPs. Our main results are a moment duality between the block-counting process of a multitype Λ-coalescent and a multitype Λ-Wright-Fisher frequency process, and a homeomorphism between the (parameter) space of multitype Λ-coalescents and that of multitype CSBPs. Moreover we construct a discrete approximation of the frequency process of a multitype CSBP that converges to the frequency process appearing in the duality result. This ``sequential sampling’’ approximation is one of our main tools in the proofs. With these results, we can in particular define ``genuinely multitype’’ Beta-coalescents.

Toth - The Pólya Web

The Pólya Walk is an up-right random walk on the discrete quadrant ℕ × ℕ which mimics the trajectory of the simple Pólya Urn. The Pólya Web is the joint realization of independent and coalescing Pólya Walks starting from all points of the discrete quadrant. We consider and partly answer various natural questions related to this object. Based on joint work (in progress) with Balázs Bárány, Piotr Śniady and Ákos Urbán.

Eckmann - Bottlenecks in Sierpinski Sandpiles

This is work in progress with Tatiana Smirnova-Nagnibeda. We consider avalanches in finite Sierpinski lattices. While the power law distributions of (large) avalanche sizes are fairly well understood, we wondered about the somewhat irregular distribution of short avalanche sizes. I think we "understand" what is happening, but proofs are missing.

Dembo - Disordered Gibbs measures and Gaussian conditioning

Given a Gaussian energy function H(x) and a random process F(x) defined on the same configuration space, what is the law of F(y) for y sampled from the disordered Gibbs measure associated with H(x)? In joint work with Eliran Subag, we resolve this at the high-temperature phase in a general setting, and in the more challenging low-temperature phase for spherical spin glasses. The answer is given in terms of the law of F(x) for deterministic x, conditional on an appropriate event. In the former case the conditioning involves only the value of H(x) at the same point x, while in the latter, we must specify also the energy and its derivatives over a sequence of critical points of certain geometry (which is induced by the model's type of replica-symmetry-breaking).

Nakajima - Recent Progress on the Ising Perceptron

The Ising perceptron is a fundamental model in statistical learning and spin glass theory. In this talk, I will mainly discuss mathematical progress on this model. As a recent development, I will present recent results on the uniqueness of the replica symmetric saddle point. 

Zeitouni - 2D polymers and log-correlated fields

I will review some recent work concerning the extrema of partition functions of 2D polymers in random environments, in the subcritical and critical regimes. Joint work with Clement Cosco and Shuta Nakajima.

Sznitman - Effective energy-enstrophy diffusion and inviscid condensation

We use Gaussian measure on R^N to define a stationary diffusion process on a cone of R². This diffusion is the inviscid limit of the laws of the “enstrophy-energy” process of an N-dimensional Galerkin-Navier-Stokes type evolution with Brownian forcing and random stirring, regardless of the strength of the stirring. With the help of this two-dimensional diffusion process, we infer quantitative condensation bounds, which for suitable forcings show an attrition of all but the lowest modes in the inviscid limit. Based on joint work with Klaus Widmayer (U. Zurich).

den Hollander - Challenges in co-evolution on networks

In this talk I present an overview of what is known and is not known about random processes on dynamic random graphs in co-evolution, i.e., with mutual feedback. The literature offers plenty of
heuristics, simulations and conjectures, but so far mathematical results are extremely scarce. I describe some of these results, with a focus on opinion dynamics. In particular, I consider random
graphs in which vertices can have one of two possible opinions. Pairs of vertices connected by an edge share their opinions according to a rate that may depend on the size of the graph. Each edge
turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same opinion or not. I exhibit joint evolution equations and describe the occurrence of phase
transitions between consensus and polarisation.

Kozma - Once Reinforced Random Walk in high dimensions

Once reinforced random walk is the simplest imaginable self-interacting random walk. The walker has a slight preference to edges it visited before, regardless of how many times it visited them. A conjecture of Vladas Sidoravicius is that in dimension d>2 the process has a phase transition in the strength of the reinforcement. We will discuss a recent result that in d≥6 the process has a transient phase. Joint work with Dor Elboim.

de Tilliere - Classification of (non)-frustrated 2D Ising models in genus 1

The goal of this talk, based on a joint work with Lucas Rey, is to prove a full classification of 2D-Ising models defined on isoradial graphs, frustrated or not, whose underlying spectral curve has genus 1. As a specific case, we recover Baxter's Z-invariant Ising model, thus extending his class of models to real coupling constants. In doing so, we identify the different behaviors of phase transition observed in the physics literature, using the underlying spectral curve. While in the case of positive coupling constants the curve is known to be Harnack, we here identify two kinds of non-Harnack curves that occur in frustrated Ising models. We prove that in all cases the curve is maximal, and undergoes an algebraic phase transition, in the sense that the genus goes from 1 to 0, thus shedding light on the physics phase transition. Our results also provide a natural framework for a further systematic study of the frustrated Ising model, amenable to proving local formulas. Note that in the course of the talk, we will define isoradial graphs and spectral curves.

van den Berg - On the torsion function for simply connected, open sets in ℝ²

For a non-empty, open set Ω ⊂ ℝ², let λ(Ω) denote the bottom of the spectrum of the Dirichlet Laplacian acting in L²(Ω). Let w_Ω be the torsion function for Ω, and let ∥·∥_p denote the L^p norm. It is shown there exists η > 0 such that ∥w_Ω∥_∞ λ(Ω) ≥ 1 + η for any non-empty, open, simply connected set Ω ⊂ ℝ² with λ(Ω) > 0. Moreover, if in addition the measure |Ω| of Ω is finite, then ∥w_Ω∥₁ λ(Ω) ≤ (1 - η)|Ω|. Joint work with Dorin Bucur, Université de Savoie.

Bovier - Using Feynman-Kac for systems of non-linear reaction-diffusion equations

Almost 50 years ago, Murray Bramson showed how the use of the Feynman-Kac representation for linear PDEs can also give very precise results on the analysis of non-linear PDEs, notably the F-KPP equation. In this talk, I will discuss the possibility of using Feynman-Kac also for systems of coupled F-KPP equations. In a particular case, I will show that this can not only provide nice results on the behaviour of wave speeds, but also very intuitive explanations. I will also discuss some ongoing work and open problems. This is joint work with Lisa Hartung.
 

Cipriani - Towards a phase transition for the integer-valued membrane model

In this talk we study the integer-valued membrane model on a finite box of the integer lattice at inverse temperature \beta. This is a random interface with bilaplacian energy constrained to the integers. For any temperature and any test function, the model has sub-Gaussian upper bounds on the tails. In dimensions 2\le d\le 4 we prove matching lower bounds for a broad class of localized test functions in a high-temperature regime in which the inverse temperature is allowed to depend on the box size and as such we identify a smallness condition on the inverse temperature under which the lower bound holds. Joint work with B. Dan (IISER Kolkata) and R. S. Hazra (Leiden University).

• David Belius, UniDistance Suisse
• Giuseppe Genovese, Albert Ludwigs Universität Freiburg
• Chiara Saffirio, Albert Ludwigs Universität Freiburg and Universität Basel

The workshop is supported by :

• SERI (Swiss State Secretariat for Education, Research and Innovation)
• Swiss Mathematical Society
• Institute of Mathematics, University of Zurich
• International Association of Mathematical Physics

Brig is well-connected to major cities in Switzerland and abroad. There are direct train connections from/to Basel, Bern, Geneva, Lausanne, Zürich, and Milano. The nearest airports are in Milano-Malpensa, Zürich, and Geneva.

Registration is required for all participants, regardless of the number of days attended.