Programme de la prochaine séance

La prochaine séance des Probabilités de Demain aura lieu mercredi 12 avril 2023 après-midi à l’Institut Henri Poincaré (IHP, 11 Rue Pierre et Marie Curie, 75005).

Programme de l'après-midi :

• 14h30-15h30 : Cristina Toninelli: Interacting particle systems with kinetic constraints
  Kinetically Constrained Models (KCM) are a class of interacting particle systems with a simple spin flip dynamics subject to local dynamical constraints. Each vertex is resampled (independently) at rate one by tossing a 1-q-coin iff a certain neighbourhood contains no particles. In other words, the holes (empty vertices) act as facilitating sites. KCM are extensively used in physics literature to model the liquid-glass transition, a longstanding open problem in condensed matter physics. Indeed, when q shrinks to 0, the presence of constraints gives rise to glassy dynamics. We will discuss techniques and results concerning the divergence of timescales for the stationary process when q shrinks to 0.
• 15h30-16h00 : Sonia Velasco, (MAP5): Hydrodynamic and hydrostatic limit for a generalized contact process with mixed boundary conditions
  Nous considérons un système de particules en interaction qui modélise la technique de l'insecte stérile en volume fini et en contact avec des réservoirs lents. Nous prouvons que la limite hydrodynamique est un système d'équations de réaction-diffusion couplées avec des conditions aux limites mixtes, qui dépendent des taux de ralentissement des réservoirs. Le but de cette présentation est de prouver l'existence d'une limite hydrostatique pour une classe de paramètres régissant la dynamique. Partant d'un équilibre microscopique, le système converge vers un équilibre macroscopique associé à l'unique solution stationnaire de l'équation hydrodynamique.
• Pause
• 16h30-17h00 : Nicolas Bouchot, (LPSM): Localization of a polymer in a random environment - Bernoulli & Interlacements

A polymer is a long chain of small molecules that is traditionally modelized using a Gibbs tranformation of the simple random walk law on Z^d, which we call the polymer measure. The Gibbs weight favors configurations that minimize a Hamiltonian, which reflects how the polymer interacts with its environement. We are interested in the typical configurations of polymers when its length goes to infinity. I will be focusing on a polymer placed in an averaged inhomogeneous Bernoulli-type percolation in Z, which can be seen as a random walk "randomly penalized" by its range. When the energy cost is i.i.d. at each site, the location of the edges of the polymer at different scales are determined through random variational problems. Then, I will explain how we hope to study a polymer in the random interlacement using recent works by Ding & Xu on the walk killed by Bernoulli percolation and works by van den Berg, Bolthausen & den Hollander on the Wiener sausage
• 17h00-17h30 : Meltem Ünel , (LMO): Critical behavior of loop models on causal triangulations
  We introduce a dense and a dilute loop model on causal dynamical triangulations. Both models are characterised by a geometric coupling constant $g$ and a loop parameter $\alpha. The dense loop model can easily be mapped onto a solvable planar tree model, whose partition function we compute explicitly and use to determine the critical behaviour of the loop model. In the dilute case, although mapping onto a planar tree model is still possible, a closed-form expression for the corresponding partition function is not obtainable using the standard methods as in the dense case. Instead, we combine the bounds obtained from the corresponding planar tree model with transfer matrix techniques to examine the critical behaviour for $\alpha$ small.