Dans cet exposé, nous introduirons certaines chaînes de Markov simples, dites “montantes-descendantes”, sur les permutations et les graphes. Une étape de la chaîne consiste à dupliquer un élément aléatoire de la permutation ou un sommet aléatoire du graphe (pas montant), puis à supprimer un autre élément/sommet aléatoire (pas descendant). Nous prouvons que ces chaînes convergent dans la limite des grandes tailles et après renormalisation du temps vers une diffusion de Feller sur l'espace des permutons et des graphons, respectivement. Nous obtenons également une formule explicite pour la distance de séparation entre la distribution des chaînes après n pas, excluant l'apparition d'un phénomène de “cut-off”. Notre approche fonctionne dans un cadre plus général : il est basé sur des relations de commutation entre les opérateurs des pas montants et descendants, et s'inspire des travaux de Fulman, Olshanski et Borodin–Olshanski sur l'espace des partitions et le simplex de Thoma. Je ne supposerai aucune connaissance préalable des permutons, graphons, diffusions de Feller, distances de séparation, seuils, ... Travail joint (et encore en cours) avec Kelvin Rivera-Lopez, Gonzaga University.[-]

These lectures review the physical and mathematical foundations of the phenomenon of spontaneous stochasticity. The essence is that stochastic classical equations and/or with randomness in their initial data, when considered along a sequence where the randomness vanishes but also the dynamics becomes singular, can have limits described by a probability measure over the non-unique weak solutions of the limiting deterministic dynamics with deterministic initial data. Furthermore, the limiting probability measure is often universal, independent of the precise sequence considered, so that the stochastic limit is then the well-posed solution of the Cauchy problem for the limiting deterministic dynamics. In the firstlecture, we discuss Lagrangian spontaneous stochasticity, which has its origin in the 1926 paper of Lewis Fry Richardson on turbulent 2-particle dispersion. As first realized by Krzysztof Gawędzki and collaborators in 1997, Lagrangian spontaneous stochasticity is necessary for anomalous dissipation of a scalar advected by a turbulent fluid flow. In the second lecture, we discuss Eulerian spontaneous stochasticity, which was anticipated in the 1969 work of Edward Lorenz on predictability of turbulent flows. After the convex integration studies of De Lellis, Székelyhidi, and others showed that Euler equations with suitable initial data may admit infinitely many, non-unique admissible weak solutions, it became clear that Lorenz' pioneering work could be understood in the framework of spontaneous stochasticity. Finally, in the third lecture we discuss outstanding problems and more recent work on spontaneous stochasticity, both Lagrangian and Eulerian. We focus in particular on statistical-mechanical analogies, on the chaotic dynamical properties necessary to achieve universality,on the use of renormalization group methods to calculate spontaneous statistics in dynamics with scale symmetries, and finally on the challenge of observing spontaneous stochasticity in laboratory experiments.[-]

Complete wetting in the context of the low temperature two-dimensional Ising model is characterized by creation of a mesoscopic size layer of the "-" phase above an active substrate. Adding a small positive magnetic field h makes "-"-phase unstable, and the layer becomes only microscopically thick. Critical prewetting corresponds to a continuous divergence of this layer as h tends to zero. There is a conjectured 1/3 (diffusive) scaling leading to Ferrari-Spohn diffusions. Rigorous results were established for polymer models of random and self-avoiding walks under vanishing area tilts.
A similar 1/3-scaling is conjectured to hold for top level lines of low temperature SOS-type interfaces in three dimensions. In the latter case, the effective local structure is that of ordered walks, again under area tilts. The conjectured scaling limits (rigorously established in the random walk context) are ordered diffusions driven by Airy Slatter determinants.
Based on joint walks with Senya Shlosman, Yvan Velenik and Vitali Wachtel.[-]

We first introduce the Metropolis-Hastings algorithm. We then consider the Random Walk Metropolis algorithm on $R^n$ with Gaussian proposals, and when the target probability measure is the $n$-fold product of a one dimensional law. It is well-known that, in the limit $n$ tends to infinity, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension $n$, a diffusive limit is obtained for each component of the Markov chain. We generalize this result when the initial distribution is not the target probability measure. The obtained diffusive limit is the solution to a stochastic differential equation nonlinear in the sense of McKean. We prove convergence to equilibrium for this equation. We discuss practical counterparts in order to optimize the variance of the proposal distribution to accelerate convergence to equilibrium. Our analysis confirms the interest of the constant acceptance rate strategy (with acceptance rate between 1/4 and 1/3).[-]

Reaction diffusion equations have been introduced during the early 20th century to model the density of populations undergoing range expansions in various contexts. These equations commonly admit travelling wave solutions, i.e. the population expands at a constant speed with a stationary profile. These deterministic models can be obtained as rescaling limits of stochastic population models when the population density tends to infinity. But do these stochastic models also admit such (random) travelling fronts? If so, what is the asymptotic speed of these fronts, and how does the nature of the front affect this speed? These questions have been the subject of many studies in the case of the Fisher-Kolmogorov-Petrovsky-Piskunov equation, and in this talk I will give some partial answers in the case of reaction-diffusion equations with a bistable reaction term.
The latter type of equations arises when one is interested in the motion of hybrid zones or the expansion of populations with an Allee effect. We shall see that their behaviour is in sharp contrast with that of the stochastic F-KPP equation.
joint work with Alison Etheridge and Sarah Penington[-]

We study a class of individual-based, fixed-population size epidemic models under general assumptions, e.g., heterogeneous contact rates encapsulating changes in behavior and/or enforcement of control measures. We show that the large-population dynamics are deterministic and relate to the Kermack-McKendrick PDE. Our assumptions are minimalistic in the sense that the only important requirement is that the basic reproduction number of the epidemic $R_0$ be finite, and allow us to tackle both Markovian and non-Markovian dynamics. The novelty of our approach is to study the "infection graph" of the population. We show local convergence of this random graph to a Poisson (Galton-Watson) marked tree, recovering Markovian backward-in-time dynamics in the limit as we trace back the transmission chain leading to a focal infection. This effectively models the process of contact tracing in a large population. It is expressed in terms of the Doob h-transform of a certain renewal process encoding the time of infection along the chain. Our results provide a mathematical formulation relating a fundamental epidemiological quantity, the generation time distribution, to the successive time of infections along this transmission chain.[-]

The main purpose of this work is to provide a framework for proving that, given a family of random maps known to converge in the Gromov--Hausdorff sense, then some (suitable) conditional families of random maps converge to the same limit. As a proof of concept, we show that quadrangulations with a simple boundary converge to the Brownian disk. More precisely, we fix a sequence $(p_n)$ of even positive integers with $p_n\sim2\alpha \sqrt{2n}$ for some $\alpha\in(0,\infty)$. Then, for the Gromov--Hausdorff topology, a quadrangulation with a simple boundary uniformly sampled among those with $n$ inner faces and boundary length $p_n$ weakly converges, in the usual scaling $n^{-1/4}$, toward the Brownian disk of perimeter $3\alpha$.[-]