Workshop on Random walks, localisation and reinforcement

Paris 6, Jussieu, salle Paul Lévy (corridor 16-26, room 209)
June 12-14, 2023

Alejandro Ramírez (Santiago, Pontificia Universidad Católica de Chile) Ballisticity, large deviations and KPZ universality for random walks in random environment
Abstract: We will present the model of random walk in random environment on the hypercubic lattice emphasizing three aspects of it: Class 1. Definitions and the directional transience-ballisticity conjecture; Class 2. Large deviations. Class 3. KPZ universality, intermediate disorder and exact solvability.

Daniel Kious (Bath) Random walk on the simple symmetric exclusion process
Abstract: In this talk, I will overview three works on random walks in dynamical random environments, in collaboration with Hilário and Teixeira for the first one, with Conchon–Kerjan and Rodriguez for the second, and Baldasso, Hilário and Teixeira for the third. Our main interest is to investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, with density ρ ∈ [0, 1] and rate γ > 0. 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. Two of the papers we will discuss imlpy a law of large numbers for the random walker for all densities ρ except at most one value ρ0, where the speed, seen as a function of the density, possibly jumps to, from or over 0. Second, we prove that, for any density corresponding to a non-zero speed regime, the fluctuations are diffusive and a Central Limit Theorem holds. I will mention a third paper, where we consider an environment with better mixing properties and investigate the fluctuations of this model in the zero-speed regime, via Russo-Seymour-Welsh techniques.

Alexandre Legrand (Toulouse) Adsorption transition at the surface of a self-avoiding, collapsed polymer
Abstract: The adsorption phase transition of a polymer to an attractive hard wall is a well-known phenomenon, and has been studied extensively in the mathematics literature: if the attraction intensity to the wall is larger than some critical value, the polymer gets localized in the vicinity of the wall, otherwise it wanders away. In this talk we additionally consider that the polymer is dipped in a poor solvent, with which it interacts repulsively. If the repulsion intensity is large, the polymer folds over on itself into a compact globule, minimizing its interactions with the solvent. We prove that in this "collapsed" regime, the polymer still undergoes the aforementioned adsorption transition. However, only the outter, bottommost layer of the globule may adhere to the wall: therefore this is not a proper phase transition, and does not appear in the free energy; this phenomenon is called a "surface transition", and is proven by deriving sharp asymptotics of the partition function of this model.

Chiranjib Mukherjee (Muenster) The continuous directed polymer in $d\geq 3$ as a Gaussian multiplicative chaos
Abstract: We construct the continuous directed polymer measure in $d\geq 3$ in the entire weak disorder regime, and identify it as a Gaussian multiplicative chaos, in the sense of Shamov, but now on path spaces. We show that it is supported on thick paths and study the geometric properties of the measure via its moments and Hölder exponents. Joint work with Rodrigo Bazaes (Münster) and Isabel Lammers (Münster).

Rémy Poudevigne-Auboiron (University of Cambridge) Slow phase transition for the VRJP on the tree
Abstract: Joint work with Peter Wildemann (Cambridge). The vertex reinforced jump process (VRJP) is closely linked to the edge-reinforced random walk, one of the simplest and oldest example of a reinforced random walk. On Z^d it is known that the VRJP goes from a recurrent phase to a transient phase but the behaviour around the critical point is not well understood. Here we look at the behaviour of the VRJP around the critical point on the d-ary tree (in the transient phase) for some quantities like the time spent at the origin and compute the speed of the phase transition. This result is based on a link between the VRJP on trees and branching random walk.

Roberto Viveros (IME University of São Paulo) Directed Polymer for very heavy tailed random walks
Abstract: In the literature on directed polymers in random environments, two notions of strong disorder emerged: the first one is based on whether or not the partition function converges to zero and is known as strong disorder. The second depends on whether the Lyapunov exponent associated with the partition function is negative or not and is known as very strong disorder. Each of these notions corresponds to a critical temperature. Very strong disorder implies strong disorder and a conjecture for directed polymers is that these two critical points coincide. However, we will present special cases for which there is not a very strong disorder phase, while strong disorder is maintained at sufficiently low temperature. This shows that the conjecture cannot be held to complete generality.