• 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.
  • 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.
  • 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.
  • 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