Activated Random Walks, DLA, and related topics

16, 17, and 18th of MARCH 2015

Maison des Astronomes, 2 Place Le Verrier.



This workshop gathers a small group of mathematicians around a mini-course of Leo Rolla on activated random walks.



Monday: 10:00-12:00 Leo R. I, Lunch, 14:30-17:30 (3 talks)
Tuesday: 10:00-12:00 Leo R. II, Lunch, 14:30-17:30 (3 talks)
Wednesday: 10:00-12:00 Leo III, Lunch and END.

Hubert Lacoin
A mathematical perspective on metastable wetting (joint work with A. Texeira).

Summary: We investigate the dynamical behavior of an interface or polymer, in interaction with a distant attractive substrate. The interface is modeled by the graph of a nearest neighbor path with non-negative integer coordinates, and the equilibrium measure associates to each path $\eta$ a probability proportional to $\lambda^{H(\eta)}$ where $\lambda$ is non-negative and $H(\eta)$ is the number of contacts between $\eta$ and the substrate. The dynamics is the natural "spin flip" dynamics associated to this equilibrium measure. We let the distance to the substrate at both polymer ends be equal to $aN$ where $0 < a < 1/2$ is a fixed parameter, and $N$ is the length the system. With this setup, we show that the dynamical behavior of the system crucially depends on $\lambda$: when $\lambda \leq 2/(1-2a)$ we show that the system only needs a time which is polynomial in N to reach its equilibrium state, whereas if $\lambda > 2/(1-2a)$ the mixing time is exponential in $N$ and the system relaxes in an exponential manner which is typical of metastability.

Tal Orenshtein
Excited Mob (joint work with Gideon Amir).

Summary: Consider two generalizations of a one dimensional Excited random walk: A) k Cookie walkers are placed on to the same cookie environment, and walk according to some scheduling. (We call this "k-mob walk") B) There are k Cookie walkers, each one in turn walks on the (j-1)-time leftover environment, j=1,...,k. In the talk we shall present the following result, and discuss ideas from its proof. Under standard assumptions, namely that the cookie environment is i.i.d., elliptic and bounded, we have in both cases A) and B) that (i) all the walkers are transient to the right a.s. if and only if $\delta>k$, and (ii) they all have positive speed if and only if $\delta>k+1$, where $\delta$ is the expected drift per site. In particular, when $\delta>3$, the walker on the (1st) leftover environment is an example of a ballistic Cookie walk with cookie distribution which is not a trivial modification of an i.i.d. one (but have stationary and ergodic properties). A fundamental ingredient of the proof is an "Abelian" property of the deterministic cookie model (where the walkers follow pregiven instructions, or "arrows", rather than cookies)

Bruno Schapira
On a simple model of balanced self-interacting walk (joint with Y. Peres and P. Sousi and G. Kozma and I. Benjamini).

Summary: We will be interested in the following model, originally introduced by Itai Benjamini. Consider the process on the two-dimensional grid, which at first visit to a site either jumps up or down, and at further visits either jumps to the right or to the left (always with probability 1/2 for each possibility). The question is to know whether the process is recurrent or not. We will not answer this question here, since it is still open, but instead we will be interested in extensions of the problem in higher dimensions.

Nathanael Enriquez
Two uses of an idea of Rolla, Sidoravicius, Surgailis and Vares to recover asymptotics of longest increasing subsequences.

Summary: Let us drop in an iid manner points on the square lattice with probability p. We ask the problem of the asymptotics of the maximal number of points a non decreasing path starting at the origin and ending at (n,n) can contain. The same question is asked for increasing paths. These two questions were solved twenty years ago by T. Seppalainen in two different works. It turns out that the introduction of two stationary systems of broken lines provides a simple and unified proof of these results. The proof of stationarity is based on a local balance property which already appears in a work of Rolla, Sidoravicius, Surgailis and Vares. (Joint with Anne-Laure Basdevant, Lucas Gerin and Jean-Baptiste Gouéré)

Ariel Yadin
Harmonic measure on expanders.

Summary: In joint work with Itai Benjamini we study the harmonic measure of a set in a finite graph, and how the spectral gap (or mixing) of the random walk affect these measures. In rapidly mixing situations one could expect that the harmonic measure should be close to the uniform measure. We make this intuition precise and quantify it. In the talk I will contrast the behavior of the harmonic measure on expanders to that of the more classical Euclidean case, namely where the underlying graph is the d-dimensional torus. For example, I will demonstrate that a Makarov type theorem cannot hold for expanders. Time permitting I will talk about the connection to DLA on finite graphs.