Clément LAURENT (Université de Provence)
Ballisticity of random walk in random environment in dimension 2
 

In 2002, Sznitman introduced an effective criterion which allowed to get condition (T') and then ballisticity, assuming that the environment is iid and also uniform ellipticity. This criterion can be checked by inspection of the environment in a finite box. In 2003, Sznitman showed using this effective criterion that a random walk in random environment on Z^d whose transition probabilities are ε-perturbations of the simple random walk is ballistic if the expected local drift is bigger than ε^α with α < 5/2 in d=3 and α < 3 in d=4. Recently, Berger, Drewitz and Ramirez introduced the so-called polynomial condition which is an improvement of the Sznitman effective criterion to get condition (T'). Using this polynomial condition we show that for an ε-perturbation of the simple random walk in dimension 2, if the expected local drift is bigger than ε^2 then the random walk is ballistic.

Stein BETHUELSEN (CWI Amsterdam)
Random Walks in Dynamic Random Environment
 

In recent years increasing interest has been devoted to the study of random walks in dynamic random environment. In this talk we address the main motivation for studying such models and in particular the challenges that arrises when one considers dynamics which has strong space-time correlations. As our main example we consider a random walk on top of the supercritical contact process on Z^d. The talk is (partly) based on ongoing work together with Markus Heydenreich (Leiden/CWI) and Rob van den Berg (VU / CWI).

Renato DOS SANTOS (Université de Lyon)
A quenched central limit theorem for random walks in random scenery in dimension 2
 

Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They are easy to define but exhibit nevertheless a very rich behaviour. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field $\xi = (\xi_x)_{x \in \Z^d}$ of i.i.d.\ random variables, which is called the \emph{random scenery}, and a random walk $S = (S_n)_{n \in \N}$ evolving in $\Z^d$, independent of the scenery. The RWRS $Z = (Z_n)_{n \in \N}$ is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n \xi_{S_k}$. The law of $Z$ under the joint law of $\xi$ and $S$ is called ``annealed'', and the conditional law given $\xi$ is called ``quenched''. Recently, central limit theorems under the quenched law were proved for $Z$ by Nadine Guillotin-Plantard and Julien Poisat in dimension $d \ge 3$. We will discuss the extension of their result to $d=2$. Joint work with Nadine Guillotin-Plantard and Julien Poisat.

Xinxin CHEN (Université Paris 6)
Accessibility percolation on N-trees
 

Accessibility percolation is a new type of percolation problem inspired by evolutionary biology. We consider a regular tree such that each individual has exactly N children. To every individual of this tree, we assign an i.i.d. random variable which is uniformly distributed in [0,1]. We say that one individual is accessible if along the shortest path from the root to this individual, the assigned random variables are increasing. We only keep the accessible individuals and kill the others. We are interested in the asymptotic behavior of the evolution of the population as N goes to infinity.

Julien POISAT (Universiteit Leiden)
Large deviation principles for words drawn from correlated letter sequences
 

When a sequence of i.i.d. letters is cut into words according to i.i.d. renewal times, a sequence of i.i.d. words is obtained. In the annealed LDP (large deviation principle) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In Birkner, Greven and den Hollander (2010) the quenched LDP (= conditional on a typical letter sequence) was derived for the case where the renewal times have an algebraic tail. The rate function turned out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Later on, these LDP's were used to derive variational formulas for polymer models in random media, namely the random pinning model (Cheliotis and den Hollander; 2013) and the copolymer model (Bolthausen, den Hollander and Opoku; preprint 2012). In this talk we will first present a heuristic derivation of these LDP for words, and then show how to extend them to letter sequences that are not i.i.d. Both LDP's actually carry over when the letter sequence satisfies a mixing condition called summable variation. The rate functions are again given by specific relative entropies w.r.t. the reference law of words, respectively, letters. But since neither of these reference laws is i.i.d., several approximation arguments are needed to obtain the extension. This is joint work with Frank den Hollander.

Laurent TOURNIER (Université Paris 13)

On the greedy cleaning problem  

Consider the following problem: a tourist wishes to visit a series of landmarks (infinitely many) and decides to always go to the closest non-visited place in his list. Depending on the spatial distribution of landmarks, does this strategy succeed?, i.e. does every landmark get eventually visited? I will give several open questions and discuss a special case. This is joint work with V. Sidoravicius and L. Rolla.

Moustapha BA (Université de Provence)

Sobolev Inequality and Individual invariance Principle for diffusions in Periodic Media  

We prove here, using stochastic analysis methods; the invariance principle for a Rd- diffusions d >=2; involving in periodic potential beyond uniform boundedness assumptions. The potential is not assumed to have any regularity. So the stochastic calculus theory for processes associated to Dirichlet forms used to justify the existence and uniqueness of this process starting for almost all x 2 Rd; allows us to show an invariance principle for almost all starting point (individual invariance principle). For that, we show then one Sobolev inequality to bound the probability of transition associated to the time changed diffusion for times large enough and deduce the existence of one bounded density. This property allows us to prove the tightness of the sequence of processes in the uniform topology; the proof of the convergence in finite dimensional distribution is very standard: construction of corrector and central limit theorem for martingale with continuous time (Helland 1982). The approach used here is the same as in [5] (Mathieu 2008): the notion of time changed process by an additive functional. keywords: Sobolev inequality, invariance principle, diffusions, periodic potential.

Bruno SCHAPIRA (Université de Provence)

Vertex Reinforced Random Walk on the complete graph  

We consider the model of Vertex Reinforced Random Walk introduced by Pemantle, in the linear regime. Here we will focus on the strongly reinforced regime, more precisely when the weight sequence is of the form w(n) = n^\alpha, with \alpha > 1. We will see that on the complete graph, some curious phenomenon happens. Joint work with Michel Benaïm and Olivier Raimond.

Daniel KIOUS (Université Toulouse 3)

Stuck Walks: a conjecture of Erschler, Tóth and Werner  

In 2010, Erschler, Tóth and Werner introduced the so-called Stuck Walks, which are a class of self-interacting random walks for which there is competition between repulsion at small scale and attraction at large scale. They proved that, for any positive integer L, if the relevant parameter belongs to a certain interval, then such random walks localize on L+2 sites with positive probability. They also conjectured that it is the almost sure behavior. We prove this conjecture partially, proving that the walk localizes on L + 2 or L + 3 sites almost surely, under the same assumptions.

Olivier RAIMOND (Université Paris 10)

Stochastic flows on graphs  

We study a stochastic differential equation (SDE) on a metric directed graph, directed by a Brownian motion. This SDE is an extension of Tanaka's SDE. The solutions of this SDE are flows of kernels. We describe the solutions of this SDE and explain how to construct them.

Gabriel FARAUD (WIAS Berlin)

Connection times in large ad-hoc mobile networks  

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, i.e., they are iteratedly forwarded from participant to participant over distances ² 2R, with 2R the communication radius, until they reach the recipient. In mathematical terms, this is a dynamic continuum percolation model.

Julien BUREAUX (Université Paris 10)

Random vector partitions and asymptotic formulae  

We introduce a family of Gibbs measures on the set of integer vector partitions indexed by two parameters. When the parameters are carefully calibrated, we can prove a strong local limit theorem that yields Hardy-Ramanujan like formulae for the number of vector partitions.

Alexander FRIBERGH (Université Toulouse 3)

On the monotonicity of the speed of biased random walk on a Galton-Watson tree without leaves  

We will present different results related to the speed of biased random walks in random environments. Our focus will be on a recent paper by Ben Arous, Fribergh and Sidoravicius proving that the speed of the biased random walk on a Galton-Watson tree without leaves is increasing for high biases. This partially solves a question asked by Lyons, Pemantle and Peres.

Pierre MATHIEU (Université de Provence)

Aging for the REM  

We show aging of Glauber-type dynamics on the random energy model, in the sense that we obtain the scaling limits of the clock process and of the age process. The latter encodes the Gibbs weight of the configuration occupied by the dynamics. Both limits are expressed in terms of stable subordinators. (Joint work with J.C. Mourrat)

Pierre TARRES (University of Oxford)

To be announced