Colloquium in probability

The frog model is a classical branching model for the spread of an infection. In this model, sleeping frogs are placed on the vertices of a graph and initially one vertex is activated. Active frogs move as simple random walks and wake up all sleeping frogs they encounter. Motivated by the addition of an immunological response, we present an extension of this model in which sleeping frogs must be visited a random number of times, i.i.d. as some $I$, before they awaken, and the active frogs that attempt to wake them up are killed. We examine the propagation speed and demonstrate that the frogs spread ballistically for a $2+\varepsilon$-moment assumption on $I$, but sublinearly for a more heavy-tailed $I$. By constructing a series of renewal times, at which the front becomes independent of the past, we are able to derive a shape theorem under more technical assumptions.

Random fields of gradients on the lattice are a class of model systems arising in the study of random interfaces, random geometry, field theory and elasticity theory. The gradient fields we consider are characterised by an imposed boundary tilt and the free energy (called surface tension in the context of random interface models) as a function of tilt. Two questions of interest are whether the surface tension is strictly convex and whether the large-scale behaviour of the model is that of the massless free field (Gaussian universality class). Where the Hamiltonian of the system is determined by a strictly convex potential, good progress has been made on these questions over the last three decades. For models with non-convex energy fewer results are known. Open problems include the conjecture (verified recently in the strictly convex case) that, in any regime where the scaling limit is Gaussian, its covariance (diffusion) matrix should be given by the Hessian of surface tension as a function of tilt. I will survey some recent advances in this direction using renormalisation group arguments and describe our result confirming the conjectured behaviour of the scaling limit on the torus and in infinite volume for a class of non-convex potentials in the regime of low temperatures and small tilt. This is based on joint work with Stefan Adams.

The abelian sandpile model introduced by Per Bak, Chao Tang and Kurt Wiesenfeld was the first discovered example of a dynamical system exhibiting self-organized criticality. Since its introduction in 1987, the model has seen widespread research interest from mathematicians and physicists alike, with a focus on explaining the complex global behaviour that emerges from the interplay of the local toppling rules. In my talk, I will introduce the abelian sandpile model, its toppling rules and how we use these dynamics to define the abelian sandpile Markov chain. We will cover the most important aspects of the sandpile Markov chain, and discuss how these apply to modern research questions on the abelian sandpile model about the distribution of particles and avalanche sizes, with a focus on how the model behaves on fractal state spaces.

We consider a stochastic process on the graph $\mathds{Z}^d$. Each $x\in \mathds{Z}^d$ starts with a cluster of size 1 with probability $p \in (0,1]$ independently. Each cluster $C$ of performs a continuous time SRW with rate $\abs{C}^{-\alpha}$. If it attempts to move to a vertex occupied by another cluster, it does not move, and instead the two clusters connect via a new edge. Focusing on dimension $d=1$, we show that for $\alpha>-2$, at time $t$, the cluster size is of order $t^\frac{1}{\alpha + 2}$, and for $\alpha \le -2$ we get an infinite component. Additionally, for $\alpha = 0$ we show convergence in distribution of the scaling limit.

A sum rule is an identity connecting the entropy of a measure with the coefficients involved in the construction of its orthogonal polynomials. It is possible to prove sum rules using large deviation theory. We consider the weighted spectral measure of random matrices and prove a large deviation principle when the size of the matrix tends to infinity. The measure may be described by its spectral information or its recursion coefficients. This allows to write the rate function in two different ways, which leads to the sum rule. In this talk I present an extension to unitary random matrices in the multi-cut case, when the limit of the spectral measure is supported by several arcs of the unit circle. In this case the rate function cannot be given explicitly. We can still state a sum rule under additional conditions on the recursion coefficients, which are related to finding a specific representative in the Aleksandrov class of measures. The talk is based on a joint work with Fabrice Gamboa and Alain Rouault.

In this presentation, we will study a covering process in the discrete one-dimensional torus that uses connected arcs of random sizes. More precisely, we fix a distribution $\mu$ on $\mathbb{N}$ and for every $n\geq 1$ we will cover the torus $\mathbb{Z}/n\mathbb{Z}$ as follows: at each time step, we place an arc with a length distributed as $\mu$ and in a uniform starting point. Eventually, the space will be entirely covered by these arcs. Changing the arc length distribution $\mu$ can potentially change the limiting behavior of the covering time. In this lecture, we will expose four distinct phases for the fluctuations of the cover time in the limit. These phases can be informally described as the Gumbel phase, the compactly support phase, the pre-exponential phase, and the exponential phase. Furthermore, we expose a continuous-time cover process that works as a limiting distribution within the compactly support phase.

The Ising Model is a classical model in statistical physics describing the behavior of ferromagnetic moments on a lattice interacting via a local interaction. When the lattice is one-dimensional and in the case of homogeneous nearest-neighbor interaction, the model is known to be exactly solvable (and simple). However, the disordered version of the one-dimensional Ising Model (called the Random Field Ising Chain), where the chain interacts with an i.i.d environment, is a much more challenging model. In particular, it exhibits a pseudo-phase transition as the strength Gamma of the inner-interaction goes to infinity. A description of the typical configurations when Gamma is large has been given in the physical literature in terms of a renormalisation group fixed point. In this talk, we will present and discuss the RFIC model, on the level of the free energy and on the level of configurations. We will consider the cases of both centered and uncentered external fields. The notion of Gamma-extrema of the Brownian motion, introduced by Neveu and Pitman, will play a crucial role in our analysis.

Pattern formation phenomena are omnipresent in the natural sciences (especially chemistry, biology, and physics). Consequently, patterns, their formation, and their stability have been studied extensively in the theory of dynamical systems. However, while the deterministic side is in large parts well understood, very little is known on the stochastic side. We want to use geometric methods and techniques from dynamical systems and regularity structures to extend certain deterministic stability results to generalised patterns in singular SPDEs.

No-Free-Lunch theorems are important results in the mathematical foundations of statistical learning. They typically state that, in expectation w.r.t. a uniformly chosen target concept, no machine learning algorithm performs better on unseen data than random guessing. Put differently, one algorithm can only outperform another when being supplied with sufficient a priori knowledge by means of training data or design. In this talk, I will present a new kind of No-Free-Lunch theorem, namely for so-called autoregressive models, most prominently used in Large Language models powering, e.g., OpenAI's ChatGPT. These can be represented by higher-order Markov chains whose kernels are learned during training. I will discuss the key points of its proof and put the result into perspective to scenarios relevant to natural language processing.

In this talk, we present a physically-motivated model of diffusion dynamics for a system of n hard spheres (colloids) evolving in a bath of infinitely-many very small particles (polymers). We first show that this two-type system with reflection admits a unique strong solution. We then explore the main feature of the model: by projecting the stationary measure onto the subset of the large spheres, these now feel a new attractive short-range dynamical interaction between each other, known in the physics literature as a depletion force, due to the (hidden) small particles. We are able to construct a natural gradient system with depletion interaction, having the projected measure as its stationary measure. Finally, we will see how this dynamics yields, in the high-density limit for the small particles, a constructive dynamical approach to the famous discrete geometry problem of maximising the contact number of n identical spheres. Based on joint work with M. Fradon, J. Kern, and S. Rœlly.

The usage of Gibbs point processes to model particle systems is a well established method. One writes the measured positions of N particles (restrained in a compact of finite volume V) to be random, and the distribution depends on the set interaction between the particles of interest. The goal is then to take the thermodynamic limit (N,V->+oo) and study the limit process to deduce properties of the original physical system. In the case of bosonic systems, this procedure is not straightforword at all, especially when one adds interactions between the particles. We will present a construction of a thermodynamic limit for superstable interactions, with a DLR equation on the limit process. Although we dot not prove the existence of interlacements (which are indication of Bose-Einstein condensation) in infinite volume, the limit process is naturally a distribution over finite loops and interlacements.

We consider a problem of stochastic optimal control with separable drift uncertainty in strong formulation on a finite horizon. The drift coefficient of the state process is multiplicatively influenced by an unobservable random variable, while admissible controls are required to be adapted to the observation filtration. Choosing a control actively influences the state and information acquisition simultaneously and comes with a learning effect. The problem, initially non-Markovian, is embedded into a higher-dimensional Markovian, full information control problem with control-dependent filtration and noise. To that problem, we apply the stochastic Perron method to characterize the value function as the unique viscosity solution to the HJB equation, explicitly construct ε-optimal controls and show that the values of strong and weak formulations agree. Numerical illustrations show a significant difference between the adaptive control and the certainty equivalence control

In this talk, we will derive the main ideas to determine the hydrodynamical behaviour of the Totally Asymmetric Simple Exclusion Process (TASEP) on Z, based on a paper by P. Ferrari that leverages the microscopic properties of the process. Specifically, we will emphasize how the nearest neighbour property and total asymmetry play a crucial role in the proof. Additionally, we will discuss the key role of second-class particles in this context, noting how these objects interestingly capture the macroscopic behaviour of the system.

We study the (binary) branching random walk when the heights of all particles in the most recent generation are conditioned to be positive. We obtain sharp asymptotics for the probability of this event and for various statistics, conditional on its occurrence. In particular, we identify the repulsion profile followed by the conditional field, and derive limits in law for its maximum, minimum and associated additive martingales. These results show, among other things, that the laws of the conditional and unconditional fields are mutually singular in the limit.

Branching Brownian motion, branching random walks, and the F-KPP equation have been the subject of intensive research during the last couple of decades. By means of Feynman-Kac and McKean formulas, the understanding of the maximal particles of the former two Markov processes is related to insights into the position of the front of the solution to the F-KPP equation. We will discuss some recent result on extensions of the above models to spatially random branching rates and random nonlinearities. Interestingly, the introduction of such inhomogeneities leads to a richer and much more nuanced picture when compared to the homogeneous setting.

The dilute Curie-Weiss model is the Ising model on a (dense) Erdös-Rényi graph G(N,p). It was introduced by Bovier and Gayrard in 1990s. There the authors showed that on the level of laws of large numbers the magnetization as well as the free energy behave as they do in the usual Curie-Weiuss model (i.e. mean-field Ising model). We analyze CLTs for the these quantities and give several critical values for p at which these fluctuaions change. This is joint work with Zakhar Kabluchko and Kristina Schubert.

We study the contact process on scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where the neighbourhood of each vertex is updated with a rate depending on its strength. We identify the full phase diagram of metastability exponents in dependence on the tail exponent of the degree distribution and the rate of updating. The talk is based on joint work with Emmanuel Jacob (Lyon) and Amitai Linker (Santiago de Chile).

Understanding the emergence of genetic diversity patterns in expanding populations is of longstanding interest in population genetics. In this talk, I will introduce a model that can be used to gain some insight on the evolution of genetic diversity patterns at the front edge of an expanding population. This model, called the ∞-parent spatial Λ-Fleming Viot process (or ∞-parent SLFV), is characterized by an "event-based" reproduction dynamics that makes it possible to control local reproduction rates and to study populations living in unbounded regions. I will present what is currently known of the growth properties of this process, and what are the implications of these results in terms of genetic diversity at the front edge. Based on a joint work with Amandine Véber (MAP5, Univ. Paris Cité) and Matt Roberts (Univ. Bath).

In this work, we consider a modification of the usual Branching Random Walk (BRW), where the position of each particle at the last generation 𝑛 is modified by an i.i.d. copy of a random variable 𝑌, which may differ from the driving increment distribution. This model was introduced by Bandyopadhyay and Ghosh (2021) and they termed it as Last Progeny Modified Branching Random Walk (LPM-BRW). Depending on the asymptotic properties of the tail of 𝑌, we describe the asymptotic behaviour of the extremal process of this model as 𝑛 → ∞.

Maker-Breaker is a two player game performed on a graph, in which Breaker tries to cut off a special vertex (e.g. origin or root) by erasing edges while Maker tries to prevent that by fixing them. In this talk we consider the game to be played on supercritical Galton-Watson trees and determine the corresponding winning probabilities given different information regimes.

I'll introduce a certain generalization of a martingale with the following property: at each time, the conditional expectation of a future value given the past, is a weighted average of all the values comprising the past. We'll assume only that more recent values are weighted no less than older values. We'll discuss motivations and constructions, and conditions under which martingale-like behaviors, such as maximal inequalities and convergence, are present in an appropriate form.

This talk is concerned with the following epidemic process: A set of nodes is partitioned into three states: susceptible, infectious, and recovered. We start with a single infectious node. Proceeding in rounds whose length is antiproportional to the population size, a fixed amount of nodes are drawn independently at random. If at least one of the selected nodes is infectious, every susceptible node in the sample becomes infected. Moreover, any infectious vertex recovers independently at a constant rate. If the expected amount of infections caused by single node is less than one, the epidemic dies out quickly and leaves almost the entire population untouched. If it is above one, either the infection dies out quickly or a large outbreak occurs, during which a non-vanishing fraction of the population is affected. Moreover, if enough nodes are infectious at the same time, the system’s behaviour is essentially deterministic.