Exposés

Angelo Abächerli, ETH Zürich

Title: Level-set percolation of the Gaussian free field on large d-regular expanders.

Abstract: In this joint project with Jiří Černý we study level-set percolation of the zero-average Gaussian free field on a class of large d-regular graphs with d larger equal 3, containing d-regular expanders of large girth and typical realisations of random d-regular graphs. Through suitable local approximations of the zero-average Gaussian free field by the Gaussian free field on the infinite d-regular tree we are able to establish a phase transition for level-set percolation of the zero-average Gaussian free field which occurs at the critical value for level-set percolation in the infinite model.

Tianyi Bai, LAGA, Paris 13

Title: Cover time on trees.

Abstract: Cover time is the time required for a random walk to visit all nodes in a graph, and we study the cover time for the simple random walk and the λ-biased walk on trees. We will give a scaling limit by studying the extremal landscape of the corresponding discrete Gaussian free field and local times.

Xinxin Chen, Université Lyon 1 (ICJ)

Title: Edge local times of randomly biased random walk on trees.

Abstract: We consider a recurrent random walk on trees for which the environment is given by a branching random walk. In the diffusive or sub-diffusive case, we study the edge local times up to the $n$-th return to the root and obtain, under the annealed and quenched probability, the asymptotical behaviours of the largest edge local time and that of the number of edges visited at least $n^\theta$ times, as well as the effective conductance of the corresponding electrical network.

Thomas Gérard, Université Lyon 1

Titre : Représentations du VRJP comme mélange de processus de Markov et frontière de Martin.

Résumé : Le processus de saut renforcé par sommets (VRJP) sur un graphe pondéré peut s'écrire comme un mélange de processus de Markov, autrement dit comme une marche aléatoire en milieu aléatoire. Cette représentation peut être construite en utilisant un certain opérateur de Schrödinger aléatoire. Cet exposé concerne la classification des représentations du VRJP. On montrera qu'une telle représentation fait toujours intervenir le même opérateur de Schrödinger, ainsi qu'une fonction positive harmonique pour cet opérateur. En utilisant la notion de frontière de Martin, on pourra alors donner une forme plus générale pour toutes les représentations. Ce travail s'appuie sur des résultats de C. Sabot, P. Tarrès et X. Zeng.

Daniel Kious, University of Bath

Title: Random walk on the simple symmetric exclusion process.

Abstract: In a joint work with Marcelo R. Hilário and Augusto Teixeira, we investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density $\rho \in [0, 1]$ of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities $\rho$ except for at most two values $\rho_−, \rho_+ \in [0, 1]$, where the speed (as a function of the density) possibly jumps from, or to, 0. Second, we prove that, for any density corresponding to a non-zero speed regime, a Central Limit Theorem (CLT) holds. For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. Finally, we prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

Titus Lupu, CNRS

Title: A construction of reinforced and self-repelling diffusions on R.

Abstract: In dimension 1 we construct continuous self-interacting stochastic processes out of Bass-Burdzy flows. We also show that these processes are fine mesh limits of discret space self-interacting random walks, in particular of the Vertex Reinforced Jump Process (VRJP) and Edge Reinforced Random Walk (ERRW). This is joint work with Christophe Sabot et Pierre Tarrès.

Cécile Mailler, University of Bath

Title: Stochastic approximation on non-compact measure spaces and applications to infinitely-many-colour Pólya urns.

Abstract: Measure-valued Pólya processes (MVPPs) are the generalization of Pólya urns to infnitely-many colours. In this joint work with Denis Villemonais (Nancy), we use stochastic approximation techniques to prove strong convergence of MVPPs to the quasi-stationnary distribution of a Markov process with absorption (that depends on the MVPP considered). The main difficulty is that we allow our set of colours to be non-compact and thus need to consider a stochastic approximation taking values in the set of measures on a non-compact space; we use Lyapunov functions to treat this non-compact case.

Maximilian Nitzschner, ETH Zürich

Title: Disconnection in two percolation models with strong correlations.

Abstract: This talk deals with disconnection problems in two strongly correlated percolation models, namely level-sets of the Gaussian free field (GFF) and the vacant set of random interlacements, both in dimensions larger or equal than three. More specifically, we study the 'disconnection event' that either the GFF level-set below a given level, or random interlacements, disconnect the discrete blow-up of a compact set from the boundary of an enclosing box. We give asymptotic bounds on the probability of this event in a strongly percolative regime. Moreover we present results on the behaviour of local averages of either the GFF, or occupation times of random interlacements under disconnection, that may be seen as an instance of entropic repulsion. This talk is based on joint work with Alberto Chiarini.

Julien Poisat, Ceremade/ Université Paris-Dauphine

Title: A Limit Theorem for the Survival Probability of a One-Dimensional Simple Random Walk among Power-Law Renewal Obstacles.

Abstract: We consider a one-dimensional simple random walk surviving among a field of static soft obstacles : each time it meets one of them the walk is killed with a fixed probability. The positions of the obstacles are sampled independently from the walk and according to a renewal process. Moreover, the increments between consecutive obstacles, or gaps, are assumed to have a power-law decaying tail. We prove convergence in law for the properly rescaled logarithm of the quenched survival probability, as time goes to infinity. The limiting law writes as a variational formula with both universal and non-universal features. The latter involves (i) a Poisson point process that emerges as the universal scaling limit of the properly rescaled gaps and (ii) a quantity that we call "asymptotic cost of crossing per obstacle" and that may, in principle, depend on the details of the gap distribution. Our proof suggests a confinement strategy of the walk in a single large gap.

This model may also be seen as a (1+1)-directed polymer among many repulsive interfaces, in which case the survival probability corresponds to the partition function and its logarithm to the finite-volume free energy.

Joint work with François Simenhaus (CEREMADE).

Rémy Poudevigne-Auboiron, Université Lyon 1 (ICJ)

Title: Monotonicity and phase transition for the Vertex Reinforced Jump Process and the Edge-Reinforced Random Walk

Abstract: We will present the Edge-Reinforced Random Walk(ERRW), the Vertex Reinforced Jump Process(VRJP) and how the two models are linked. Then we will introduce a couple of quantities relevant to the VRJP: the beta-field and the psi-field. We will then show that the psi-field for different initial weights can, under some condition, be coupled. This in turn will induce monotonicity in the model which will allow us to show that the probability that the ERRW and the VRJP are transient is non-decreasing in the initial weights.

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