Plenary Talks

Inés Armendáriz

The Box Ball System The Box Ball System, or BBS for short, was introduced by Takahashi and Satsuma in 1990 as a cellular automaton that exhibits solitons (travelling waves). In a recent work, Ferrari, Nguyen, Rolla and Wang propose a hierarchical decomposition of a fixed configuration of the BBS in solitons, called the slot decomposition, and Ferrari and Gabrielli identified the distribution of this decomposition for a random walk with negative drift. In this project we extend these results to a Brownian motion with negative drift. We consider the excursions over past minima of the trajectory, and show that they can be decomposed as a superposition of solitons. These are distributed as a Poisson process in the first quadrant of the plane, with an intensity that is homogeneous in the abscissa (associated to the location of the solitons) but not in the ordinate (denoting the size of the solitons). Joint work with Pablo Blanc, Pablo Ferrari and Davide Gabrielli.

Marielle Simon

A few scaling limits results for the facilitated exclusion process I will present some recent results which have been obtained for the facilitated exclusion process, in one dimension. This stochastic lattice gas is subject to strong kinetic constraints which create a continuous phase transition to an absorbing state at a critical value of the particle density. If the microscopic dynamics is symmetric, its macroscopic behavior, under periodic boundary conditions and diffusive time scaling, is ruled by a non-linear PDE belonging to free boundary problems (or Stefan problems). One of the ingredients is to show that the system typically reaches an ergodic component in subdiffusive time. When the particle system is put in contact with reservoirs of particles (which can either destroy or inject particles at both boundaries), we observe an usual impact on the boundary values of the empirical density. The asymmetric case can also be fully treated (for the infinite particle system): in this case, the empirical density converges to the unique entropy solution to a hyperbolic Stefan problem. All these results rely, to various extent, on a mapping argument with a zero-range process, which completely fails in dimension higher than 1. I will finally discuss some open problems and questions, especially in dimension 2. Based on joint works with O. Blondel, H. Da Cunha, C. Erignoux, M. Sasada and L. Zhao.

Franco Severo

On the phase transition of Voronoi percolation Consider the Voronoi tessellation associated to a Poisson point process on \(\mathbb{R}^d\), \(d\geq2\). Then, given \(p\in[0,1]\), color each cell black independently with probability \(p\). We are interested in the connectivity properties of the set of black cells as \(p\) varies. It is known that there is a critical point \(p_c(\mathbb{R}^d)\in(0,1)\) above which this set percolates, i.e. it contains an unbounded connected component. In this talk, we prove that the supercritical behavior is very robust in sense that, for every \(p>p_c\), the unbounded component is unique in a strong sense and the size of finite components has exponential tail. As a consequence, we deduce that if the same model is defined on the d-dimensional hyperbolic space \(\mathbb{H}^d\) (and other manifolds), then the associated critical point \(p_c(\mathbb{H}^2,\lambda)\) converges to \(p_c(\mathbb{R}^d)\) as the intensity \(\lambda\) of the Poisson point process diverges. These resultas previously known only in the case \(d=2\). Based on joint works with Barbara Dembin and Ritvik Radhakrishnan.

Leonardo Rolla

Quasi-stationary distributions for subcritical population models We consider population models with reproduction (branching process, contact process, branching random walks). This model has a phase transition in terms of the distribution of the number of children. In the subcritical case, every initial distribution is attracted to the empty configuration and, in the lack of a non-trivial stationary distribution, one studies the quasi-stationary behavior of the system. In this talk we will discuss questions of existence, uniqueness and nonuniqueness of the quasi-stationary distribution, and their relation with spatial aspects of the dynamics. Joint work with Pablo Groisman and Célio Terra.

Alexandre Stauffer

Non-monotone phase transition in interacting particle systems In this talk we will discuss a class of particle systems which has a non-monotone phase transition. One example is the spread of infection with recover (SIS) among independent random walk particles, where an infected particle can only spread the infection to a susceptible particle if the total number of particles at that site is bounded by some constant K. In this model, the survival of the infection is a non-monotone function of the particle density. In the talk we will explain the techniques used to analyze monotone models and how they have to be refined to analyze non-monotone particle systems. Based on joint works with Tom Finn and Leandro Chiarini.

Gioia Carinci

Microscopic characterization of the non-equilibrium stationary states for the Harmonic model The characterization of the invariant measures for non-reversible particle systems driven out-of-equilibrium via the action of external reservoirs is typically a difficult task. This has been achieved e.g. for the well-known exclusion process. In this talk I will show a class of boundary driven zero-range models whose non-equilibrium steady state can be explicitly characterized via a probabilistic mixture of inhomogeneous product measures. This characterization of the non-equilibrium steady state allows to compute the formula for the density large deviation function predicted by Macroscopic Fluctuation Theory and to establish the additivity principle. This is from joint works with: Chiara Franceschini, Rouven Frassek, Davide Gabrielli, Cristian Giardinà, Frank Redig and Dimitrios Tsagkarogiannis.

Hubert Lacoin

Strong disorder and very strong disorder are equivalent for directed polymers The Directed Polymer in a Random Environment is a statistical mechanics model, which has been introduced (in dimension 1) as a toy model to study the interfaces of the planar Ising model with random coupling constants. The model was shortly afterwards generalized to higher dimensions. In this latter case, rather than an effective interface model, the directed polymer in a random environment can be thought of as modeling the behavior of a stretched polymer in a solution with impurities. The interest in the model model, triggered by its rich phenomenology has since then generated a plentiful literature in theoretical physics and mathematics. An important topic for the directed polymer is the so-called localization transition. This transition can be defined in terms of the asymptotic behavior of the renormalized partition function of the model. If the finite volume partition function converges to an almost surely positive limit we say that weak disorder holds. On the other hand if it converges to zero almost surely, we say that strong disorder holds. It has been proved that weak disorder implies that the distribution of the rescaled polymer converges to standard Brownian motion while some localization results have been proved under the strong disorder assumptions. Much stronger characterizations of disorder-induced localization have been obtained under the stronger assumption that the partition function converges to zero exponentially fast. This latter regime is known as the very strong disorder regime. It has long been conjectured that strong and very strong disorder are equivalent. In this talk we will sketch a proof of this conjecture (joint with Stefan Junk).

Lisa Hartung

The branching random walk conditioned on being positive We give a precise description of a binary (Gaussian) branching random walk conditioned to be positive. Interestingly, under the conditioning the branching random walk reaches a high value already after log n steps. We also obtain a description of the maximum and minimum of the conditioned branching random walk. The talk is based on joint work with M. Fels (Technion Haifa) and O. Louidor (Technion Haifa).