Schedule

After a short introduction on forest fire models, I will focus on a recent joint paper with Pierre Nolin (Hong Kong), where we study a variant of these models in which the forest is an $n\times n$ box, and ignitions come from the boundary. In particular we show that, for the case without recoveries, the probability that the center of the box is burnt tends to $0$ as $n$ tends to $\infty$. This result is not intuitively obvious, it depends heavily on (bounds for) the values of certain (near-)critical exponents for Bernoulli percolation.

Our paper also presents a (weaker) version of the above result for the case with recoveries, and points out that our methods give some more insight to a remarkable forest fire model in the half-plane studied by Graf around 2015.

The configuration model is a standard model for generating a random graph with a prescribed degree distribution. I will describe attempts to analyze a spatial version of the model, and ongoing work to revert the spatial model to the non-spatial setting.

In 1999, Benjamini, Kalai and Schramm proved that crossing probabilities are noise sensitive. Later, Schramm and Steif (2007) and Garban, Pete and Schramm (2010) obtained quantitative versions of this result, establishing the existence of exceptional for dynamical percolation. These works rely on Fourier analysis, and are restricted to Bernoulli percolation (i.e. product measure) and the independent resampling dynamics.

In this talk, we will discuss noise sensitivity for more general percolation models, and more general dynamics. Based on a recent approach to noise sensitivity with Hugo Vanneuville (that relies on geometrical arguments and not on spectral methods), we show noise sensitivity of crossing probabilities for high temperature Ising under Glauber dynamics.

Based on a joint work with Hugo Vanneuville. 

Consider a Coulomb gas of charged particles confined to a set in the complex plane. I will discuss the following question: How does the asymptotic expansion of the free energy depend on the geometry of the set, as the number of particles tends to infinity? I will focus in particular on the case of a Jordan arc but will also touch on the related problems when the set is a Jordan curve and -domain. Based joint works with K. Courteaut (NYU) and K. Johansson (KTH).

This talk is about taking stuff seriously, and revolves around two examples. The first concerns taking math seriously, going back to the 1990s and the wonderful role model who taught me to do so. The second concerns the urgent need, today in 2025, to take AI and its societal ramifications seriously.

As an example for a random walk in random environment, we study biased random walk for dynamical percolation on the d-dimensional lattice. We establish a law of large numbers and an invariance principle for this random walk using regeneration times.

Moreover, we verify that the Einstein relation holds, and we investigate the speed of the walk as a function of the bias. While for d = 1 the speed is increasing, we show that in general this fails in dimension d ≥ 2. As our main result, we establish two regimes of parameters, separated by a critical curve, such that the speed is either eventually strictly increasing or eventually strictly decreasing. This is in sharp contrast to the biased random walk on a static supercritical percolation cluster, where the speed is known to be eventually zero.

Based on joint work with Sebastian Andres, Dominik Schmid and Perla Sousi.

On a locally finite, infinite tree $T$, let $p_c(T)$ denote the critical probability for Bernoulli percolation. We prove that every positively associated, finite-range dependent percolation model on $T$ with marginals $p > p_c(T)$ must percolate. Among finite-range dependent models on trees, positive association is thus a favourable property for percolation to occur.

On general graphs of bounded degree, Liggett, Schonmann and Stacey proved that finite-range dependent percolation models with sufficiently large marginals stochastically dominate product measures. Under the additional assumption of positive association, we prove that stochastic domination actually holds for arbitrary marginals.

Studying the class of 1-independent percolation models has proven useful in bounding critical probabilities of various percolation models via renormalization. In many cases, the renormalized model is not only 1-independent but also positively associated. This motivates us to introduce the smallest parameter $p_a^+(G)$ such that every positively associated, 1-independent bond percolation model on a graph $G$ with marginals $p > p_a^+(G)$ percolates. We obtain quantitative upper and lower bounds on $p_a^+(\Z^2)$ and on $p_a^+(\Z^n)$ as $n\to \infty$, and also study the case of oriented bond percolation. In proving these results, we revisit several techniques originally developed for Bernoulli percolation, which become applicable thanks to a simple but seemingly new way of combining positive association with finite-range dependence.

This is joint work with Aurelio Sulser.

A private information retrieval (PIR) scheme allows a user to retrieve a file from a database without revealing any information on the file being requested. In a series of recent papers, coded distributed storage systems have been considered, where every file is distributed over several servers using the same linear storage code, and some unknown set (of bounded size) of the servers are observed by an adversary. After introducing the model, we will quickly move to its mathematical abstraction, review an (elementarily) algebraic construction of a family of retrieval schemes. We show that these schemes asymptotically (as the number of files grows) very quickly achieves a conjectured capacity bound, and prove this capacity bound using information theoretic inequalities under natural technical assumptions.

We will review recent progress on two related q-state models on Z^d, the noisy voter model and the Potts model, focusing on dynamical aspects of the former and on static features of the latter at low temperature. 

Based on joint works with Patrizio Caddeo, Joseph Chen and Reza Gheissari.  

Activated random walk (ARW) is a discrete, stochastic particle system which conjecturally exhibits self-organized criticality (SOC), a phenomenon which was first studied by physicists in the 1980s in the context of complex real-world systems like sandpiles, forest fires and earthquakes. I will discuss recent progress towards establishing elements of SOC for ARW, including a new universality result. 

We study O(n)-invariant quantum spin systems, via a probabilistic representation which is a loop model with weight n^#loops. We study the ground state of the quantum model in 1D, which amounts to studying infinite volume limits of the loop model in 2D. The loop model is expected to behave like other 2D loop O(n) models and the spin O(n) model at low temperature: for n\le2, there should be polynomial decay of correlations, whereas for n>2, exponential decay. Moreover the loop model should have a unique Gibbs measure for n\le2 and should break translation invariance for n>2. We prove this n>2 behaviour for n large in a wide section of the phase diagram of the model. Joint work with Jakob Björnberg.

There are some well known examples where self-similarity or hierarchy leads to noise sensitivity. Examples include the Recursive Majority function (e.g. Ben-Or, Linial 1990 or Benjamini, Kalai Schramm 1999) and critical percolation (Benjamini, Kalai, Schramm 1999, Schramm Steif 2005, Garban, Pete and Schramm 2010 etc.). In fact there is a beautiful set of lecture notes on the topic by Garban and Steif (2012). 

I will survey some of these results and discuss some new variations of the topic motivated by questions in deep learning where the underlying measure is not the uniform measure on the cube, or sometimes not even a product measure.

Based on joint works with Frederic Koehler, Han Huang and Rupert Li.

References:

Ben or and Linial (1990)

Benjamini Kalai Schramm (1999)

Schramm and Steif (2005)

Garban Pete and Schramm (2010)

Garban and Steif (2012)

Koehler Mossel (2022)

Huang Mossel (2024)

Huang Mossel (2025)

Li Mossel (2025)

I will present a survey of historic and more recent results in the classic topic of the title.  

Given randomly placed red points and blue points in space, how can they be matched up optimally in red-blue pairs, and what does the resulting matching look like?  What does optimally even mean?  And why should we care?  I’ll explore these issues, focussing on some favourite open problems, and (putative) implications for the wider world.

We consider a system of particles lined up on a finite interval in a 3-dimensional space with Coulomb interactions between the nearest and next-to-nearest neighbours. The distribution of spacings between the consecutive particles is of interest. Assuming zero external force, we show that interactions beyond the nearest ones lead to qualitatively new features of the system. In particular, the order of decay (in terms of the total number of particles) of covariances between the spacings is changed when compared with the former nearest-neighbour case. Furthermore, we discover that the covariances between spacings exhibit the antiferromagnetic property. In the course of the proof of these results, a Central Limit Theorem for dependent random variables is established, yielding as well that the fluctuations in the considered ensemble are Gaussian.

I will define an (alternative to the FK-random cluster model) Edwards-Sokal representation of the Ising model using random currents. I will then discuss a scaling limit result which implies that the continuum Ising magnetization field and the Gaussian free field are naturally coupled. To the best of our knowledge, the existence of such a coupling was not predicted before. This completes the picture of bosonization of the planar Ising model. Based on joint work in progress with Tomas Alcalde and Lorca Heeney.

This talk is a brief overview of recent progress in the study of planar directed polymer models through their Busemann functions. We start with the lattice model and, time permitting, discuss also the Kardar-Parisi-Zhang (KPZ) stochastic partial differential equation in one space dimension.  Results presented include multiplicity of polymer Gibbs measures, nonexistence of bi-infinite polymer paths, coalescence exponents, and non-unique eternal solutions of the KPZ equation for values of the conserved quantity in a random dense set.  

Consider an iid occupied/vacant configuration on the infinite 3-ary tree, and ask if there is an infinite binary vacant subtree somewhere. At the critical density (1/9 for occupation), there are already such vacant subtrees (a hybrid phase transition: discontinuous but not quite first-order). Now resample each site according to an independent Poisson process, keeping the density critical. Are there exceptional times in this dynamics when all the vacant binary subtrees get destroyed?

By a Baire category argument, the existence of exceptional times is equivalent to the set of times at which the root is not in a vacant binary subtree being everywhere dense a.s. We don't know this, but show semi-volatility: in any time interval [0,\eps), starting from the root being in a vacant binary subtree, this event changes infinitely many times with probability at least c\sqrt{\eps}.

A related side result: in standard Bernoulli percolation on regular trees at the critical density, with a stationary Brownian dynamics (say, on the binary tree, the color of each vertex is the sign of an independent Ornstein-Uhlenbeck process), the set of exceptional times with an infinite cluster has Hausdorff-dimension 1/2. This is very similar to an unpublished result of Noam Berger, and we both conjecture that on any transitive graph 1/2 is a lower bound.

Joint work with Marek Biskup and Ábel Farkas.

The magnetization field of the 2D critical Ising model can be obtained through different natural resummations of the spins sigma_x where x \in Z^2. In this talk, I will discuss how these distinct spin-grouping approaches extend to the continuum limit. This is joint work with Paul Cahen (Université Lyon 1) and Avelio Sepúlveda (Universidad de Chile).

Forsström, Gantert and Steif recently introduced a class of processes called Poisson representable processes. In this talk, we focus on the subclass of those processes which are Poisson representable with an intensity measure that concentrates on finite sets. These have the following rather simple and explicit description: Construct a random subset of Zd by independently selecting each finite subset with some probability depending on the set up to translations and taking the union of the selected sets. Our two main questions of interest are: (1) When can such a random set be described as a finitary factor of an IID process? (2) When is such a random set stochastically dominated by a non-trivial IID process? We discuss our results, which give answers to some questions raised by Forsström, Gantert and Steif.

I will survey the thermodynamic formalism, optimal results for uniqueness of g-measures (or Doeblin measures), mixing properties, and the connection to the Dyson model. I will also present some recent results with Noam Berger, Diana Conache and Anders Johansson (PTRF 2025) and with Anders Johansson and Mark Pollicott (Math Z 2025). In the end I will mention a recent result with Noam Berger, Anders Johansson and Jeff Steif on the equality of the critical inverse temperatures for the one-sided and two-sided Dyson models.

The Laplacian is a self-adjoint operator of importance in many areas of mathematics, science, and engineering, including probability. Its eigenvalues are of particular interest. We describe an explicit estimate of eigenvalues of the Laplacian associated to homogeneous spaces, that is, spaces whose group of self-isometries acts transitively. Our proof is geometric, rather than analytic. This is joint work with Chris Judge.  Our honoree will play a cameo role.

The family of beta-splitting random trees was introduced by David Aldous in 1993. The tree is a binary tree with a given number of leaves, and is constructed by recursively splitting the set of leaves into two random subsets with sizes having a distribution given by a certain formula including a parameter beta.

The "critical" case beta = -1 turns out to be particularly interesting, and it has recently attracted renewed interest by Aldous and others, including myself.

I will talk about a few of these results, in particular a representation using exchangeable random partitions of N, and, using this, an analysis of leaf height using Mellin transforms.

(Joint work with David Aldous, see arXiv:2412.09655 and Electron. J. Probab. 30, paper no. 69.)

In its classical setting, “noise sensitivity” refers to the sensitivity of the occurrence of some event, depending on a large number of independent variables, to a small perturbation of the input variables. The notion was introduced by Benjamini, Kalai, and Schramm in 1999, and has since been studied extensively in the context of percolation. In the context of spatial growth, noise sensitivity would correspond to the sensitivity of the travel time between distant points. In this talk, we present recent work establishing noise sensitivity in the context of spatial growth for the first time: noise sensitivity in last-passage percolation. Joint work with Malo Hillairet and Ekaterina Toropova.

We consider a random graph in which vertices can have one of two possible colours. Each vertex switches its colour at a rate that is proportional to the number of vertices of the other colour to which it is connected by an edge. Each edge turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same colour or not. The resulting double-dynamics is an example of co-evolution. We prove that, in the limit as the graph size tends to infinity and the graph becomes dense, the graph process converges, in a suitable path topology, to a limiting Markov process that lives on a certain subset of the space of coloured graphons. In the limit, the density of each vertex colour evolves according to a Fisher-Wright diffusion driven by the density of the edges, while the underlying edge connectivity structure evolves according to a stochastic flow whose drift depends on the densities of the two vertex colours.

Joint work with Siva Athreya (Bangalore) and Adrian Röllin (Singapore).