# Séminaire Calcul stochastique

### organisé par l'équipe Probabilités

• ##### Graph limits, common graphs and Sidorenko's conjecture

— Jan Volec

10 février 2022 - 10:45Salle de séminaires IRMA

A systematic approach to large discrete structures using analytic tools started about 15 years ago, and since then, it has attracted a substantial attention. In the first part of the talk, we describe the limit theory of dense graphs developed by Borgs, Chayes, Lovasz, Sos, Szegedy and Vesztergombi. In the second part of the talk, we focus on so-called common graphs. A given graph H is common if the number of monochromatic copies of H in every 2-edge-coloring of an n-vertex complete graph is asymptotically minimized by a random 2-edge-coloring. In 1989, Thomason disproved a conjecture of Erdos by showing that a complete graph on at least 4 vertices is not common. The existence of a common graph with chromatic number more than 3 was open until about 10 years ago, and no example of a common graph with chromatic number more than 4 has been known. In this talk, we construct for every k>4 a common graph H_k with chromatic number at least k. This is a joint work with D. Kral and F. Wei
• ##### A neutral multi-allelic Moran model: spectral elements and cutoff

— Josué Corujo

21 avril 2022 - 10:45Salle de séminaires IRMA

We will present some spectral properties related to a neutral multi-allelic Moran model, which is a finite continuous-time Markov process. For this process, it is assumed that the individuals can be of different types (among a finite set) and they interact according to two mechanisms: a mutation process where they mutate independently of each other according to an irreducible rate matrix, and a Moran type neutral reproduction process, where two individuals are uniformly chosen, one dies and the other is duplicated. During this talk we will discuss some results related to the spectral elements of the generator of this process. We will show explicit expressions for its eigenvalues in terms of the eigenvalues of the rate matrix that drives the mutation process. Our approach does not require that the mutation process be reversible, or even diagonalizable. Additionally, we will discuss some applications of these results to the study of the speed of convergence to stationarity of the Moran process with a general mutation scheme. Under some non-restrictive hypotheses, we can prove a lower bound for the mixing time of the multi-allelic Moran process. Then we focus on the case where the mutation scheme satisfies the "parent independent" condition, where (and only where) the neutral Moran model becomes reversible. In this latter case, we can go further by proving the existence of a cutoff phenomenon for the convergence to stationarity.
• ##### Annihilation balistique à trois vitesses

— Laurent Tournier

12 mai 2022 - 10:45Salle de séminaires IRMA

Dans le modèle d'annihilation balistique, des particules sont issues d'un processus de Poisson sur la droite réelle, se déplacent à vitesses constantes choisies initialement i.i.d., et s'annihilent mutuellement lors des collisions. Ce modèle a été introduit dans les années 90 en physique en alternative aux modèles classiques de réactions contrôlées par diffusion ; cependant son comportement asymptotique reste très mal compris dès qu'il y a davantage que 2 vitesses possibles. On s'intéresse ici au cas à 3 vitesses -1, 0, +1, avec distribution symétrique, et on montre en particulier qu'il se produit une transition de phases lorsque la proportion de particules immobiles dépasse 1/4, et que le modèle présente des propriétés combinatoires remarquables. Cet exposé est basé sur des travaux en collaboration avec J. Haslegrave et V. Sidoravicius.
• ##### Matsumoto-Yor and Dufresne type theorems for a random walk on positive definite matrices.

— Jonas Arista

19 mai 2022 - 10:45Salle de séminaires IRMA

The goal is to establish analogues of the geometric Pitman 2M-X theorem of Matsumoto and Yor and of the classical Dufresne identity, for a multiplicative random walk on positive definite matrices with Beta type II distributed increments. If time permits, we discuss the connections of these theorems with a more general push and block' (Markov) dynamics on triangular arrays of matrices.
• ##### The expected number of spanning trees in random graphs with given degrees

— Mikhail Isaev

18 août 2022 - 14:00Salle de séminaires IRMA

We consider a uniformly chosen random labelled graph G with n vertices and given degree sequence dvec = (d_1, d_2, ..., d_n). We derive asymptotic formulas for the expected number of spanning trees tau(G). This graph parameter is also known as the complexity of G and has connections to a wide range of topics, including the study of electrical networks, algebraic graph theory, statistical physics and number theory.

Our proofs are based on known asymptotic enumeration results of graphs with specified substructures and degrees. The main difficulty is estimating the average of exp(F(dvec, T)) over all trees on n vertices. The function F(dvec,T) is too large to allow any useful expansion of the exponential. We do this by applying the theory of exponentials of martingales. Our martingale construction is based on the Pruffer code. As an intermediate step, we derive an exact explicit formula for substructures occurrences probabilities in a random tree which appears to be new and of independent interest.

The talk is based on the following two papers.
[1] C. Greenhill, M. Isaev, M. Kwan, B.D. McKay, The average number of spanning trees in sparse graphs with given degrees, European J. Combin,
63 (2017), 6-25.

[2] C. Greenhill, M. Isaev, B.D. McKay, Subgraph counts for dense random graphs with specified degrees, Combinatorics, Probability and Computing
30 (3) (2021), 460-497.
• ##### On the number of level sets of smooth Gaussian fields

— Dmitry Belyaev

13 septembre 2022 - 10:45Salle de séminaires IRMA

The number of zeroes or, more generally, level crossings of a Gaussian process is a classical subject that goes back to the works of Kac and Rice who studied zeroes of random polynomials.  The number of zeroes or level crossings has two natural generalizations in higher dimensions. One can either look at the size of the level set or the number of connected components. The surface area of a level set could be computed in a similar way using Kac-Rice formulas. On the other hand, the number of the connected components is a non-local' quantity which is notoriously hard to work with. The law of large numbers has been established by Nazarov and Sodin about ten years ago. In this talk, we will briefly discuss their work and then discuss the recent progress in estimating the variance and deriving the central limit theorem. The talk is based on joint work with M. McAuley and S. Muirhead.
• ##### Busemann process and infinite geodesics in the directed landscape

— Ofer Busani

20 septembre 2022 - 10:45Salle de séminaires IRMA

In Last Passage Percolation (LPP) on the lattice, to each site of Z^2 we assign a random positive weight so that the weights are i.i.d. across the lattice. To any two points such that one is located north-east to the other, we look for an up-right path that maximizes the weight between the two points. Such paths are called geodesics and have been much studied in this model as well as other LPP models. One of the reasons for the interest in such objects is that they can be used as tools to study models in the KPZ universality class. An infinite geodesic in an LPP model is and infinite up right path such that its restrictions to between any to points on it is a geodesic. Since their introduction in the 90's by Newman and Hoffman, the Busemann function and the Busemann process, have been proven to be an important tool in the study of infinite geodesics in LPP as well. The directed landscape, constructed in the breakthrough work of Dauvergen, Ortman and Virag 18', is believed to be the universal scaling limit of all metric-like (LPP, FPP etc.) models in the KPZ universality class. In a recent work, Rahman and Virag showed that infinite geodesics exist in the directed landscape as well. In this talk I will discuss the construction of the Busemann process on the directed landscape and show how it can be used to obtain new results about the infinite geodesics in the directed landscape.
• ##### Scaling limit of the eigenvalues and eigenfunctions for a 1-dimensional random Schrödinger operators

— Fumihiko Nakano

11 octobre 2022 - 10:45Salle de séminaires IRMA

We consider the 1d Schrödinger operator with decaying random potential, and study the joint scaling limit of the eigenvalues and the measures associated with the corresponding eigenfunctions. As a result, we have completely different behavior depending on the decaying rate $\alpha > 0$ of the potential : the limiting measure is equal to (1) Lebesgue measure for the super-critical case ($\alpha > 1/2$), (2) a measure of which the density has power-law decay with Brownian fluctuation for critical case($\alpha=1/2$), and, (3) the delta measure with its atom being uniformly distributed for the sub-critical case($\alpha<1/2$). We also discuss the local version of this problem.
• ##### Échantillonner une distribution grâce aux processus de Markov déterministes par morceaux

— Augustin Chevallier

18 octobre 2022 - 10:45Salle de séminaires IRMA

De nouveaux algorithmes d’échantillonnages basés sur des Processus de Markov Déterministes par Morceaux ont été développés ces dernières années. Je vais présenter ici ces algorithmes, ainsi que la théorie associée. En particulier, les outils théoriques pour montrer l'invariance de la mesure cible par rapport au processus seront développés.
• ##### Sensibilité au bruit en percolation, sans outil spectral

— Hugo Vanneuville

8 novembre 2022 - 10:45Salle de séminaires IRMA

Dans cet exposé, je parlerai du phénomène de sensibilité au bruit en percolation planaire critique. Ce phénomène peut être résumé de la façon suivante : Soit A un événement de percolation à grande échelle. Savoir que A est vérifié après avoir "un petit peu bruité" la configuration de percolation ne nous dit essentiellement rien sur la configuration non bruitée. Depuis que cette notion a été introduite (en 1999, par Benjamini, Kalai et Schramm), tous les travaux qui ont été réalisés sur ce sujet se reposent sur des outils spectraux. Ces outils sont très beaux et riches, mais ont des limitations que j'exposerai brièvement. Avec Vincent Tassion, nous avons proposé une nouvelle approche, non spectrale, qui consiste à étudier des inégalités différentielles qui décrivent comment les arêtes pivots sont affectées par le bruit (les arêtes pivots sont les arêtes qui sont à la frontière de deux grands clusters). Le but de cet exposé est d'expliquer cette approche, après avoir énoncé certains résultats du domaine, tels que les théorèmes de Schramm--Steif et Garban--Pete--Schramm.
• ##### Inégalité de déviation pour des champs aléatoires à valeurs dans un espace de Banach et quelques applications

— Davide Giraudo

15 novembre 2022 - 10:45Salle de séminaires IRMA

Après avoir évoqué les inégalités de déviation pour les suites d'accroissements de martingales à valeurs réelles, nous présenterons une inégalité de déviation pour des martingales multi-indexées à valeurs dans un espace de Banach. Le fait de considérer des variables aléatoires à valeurs vectorielles s'avère non seulement utile pour mettre en oeuvre des arguments de récurrence sur la dimension des indices, mais aussi dans certaines applications à des modèles de régression.
• ##### MacMahon's master theorem: variations, proofs and applications

— Guoniu Han

22 novembre 2022 - 10:45Salle de séminaires IRMA

MacMahon's master theorem is an identity in enumerative combinatorics and linear algebra discovered by Percy MacMahon (1916). In this talk, I will present some variations, proofs and applications of MacMahon's master theorem.
• ##### The role of correlation in diffusion control ranking games

— Nabil Kazi-Tani

29 novembre 2022 - 10:45Salle de séminaires IRMA

We study Nash equilibria in 2-players continuous time stochastic differential games, where the players are allowed to control the diffusion coefficient of their state process. We consider zero-sum ranking games, in the sense that the criteria to optimize only depends on the difference of the two players state processes. We explicitly compute the players optimal strategies, depending on the correlation of the Brownian motions driving the two state equations: in particular, if the correlation coefficient is smaller than some explicit threshold, then the optimal strategies consist of strong controls, whereas if the correlation exceeds the threshold, then the optimal controls are mixed strategies. To characterize these equilibria, we rely on a relaxed formulation of the game, allowing the players to randomize their actions. This is a joint work with Stefan Ankirchner and Julian Wendt (University of Jena).
• ##### Critères de type Foster-Lyapunov pour l'étude des distributions quasi-stationnaires

— Denis Villemonais

6 décembre 2022 - 10:45Salle de séminaires IRMA

Dans ces travaux, effectués en collaboration avec Nicolas Champagnat (IECL, Nancy France), nous proposons des hypothèses pour montrer l'existence d'une distribution quasi-stationnaire pour des processus de Markov absorbés évoluant dans des espaces non-bornés. Durant l'exposé, nous aborderons les définitions et propriétés usuelles des distributions quasi-stationnaires, les critères généraux sus-mentionnés et leurs applications aux systèmes dynamiques perturbés, aux diffusions absorbées au bord d'un ouvert, au processus de Polya à valeur mesure (une collaboration avec Cécile Mailler, de l'université de Bath) et aux processus de branchement.