Let each site of the square lattice Z² be declared "closed" with probability p, a "target" with probability q, and "open" with probability 1-p-q, where 0≤p≤p+q≤1. Different sites are independent.
Consider the following game: a token starts at the origin, and the two players take turns to move it from its current site x to a site in {x+(0,1),x+(1,0)}. A player moving to a closed site loses immediately. A player moving to a target site wins immediately. Otherwise, the game continues.
Is there positive probability that the game is drawn with best play - i.e. that neither player can force a win? We show that this question is equivalent to the question of ergodicity of a certain elementary one-dimensional probabilistic cellular automaton (PCA). We prove that the PCA is ergodic whenever p+q>0, and correspondingly that the game on Z² has no draws.
On the other hand, we prove that for q=0 and p sufficiently small, the analogous game does exhibit draws on various directed graphs in dimension d at least 3. This is proved via a dimension reduction to a hard-core lattice gas in dimension d-1. We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice Z^d for d at least 3, but here our method encounters a fundamental obstacle.
If time permits, I will also mention similar games on undirected lattices, where there are a few related results but also many open questions.
This is joint work with Alexander Holroyd and Irene Marcovici.

We consider large random matrices with centered, independent entries but possibly different variances and compute the limiting distribution of eigenvalues. We then consider applications to long time asymptotics for systems of critically coupled differential equations with random coefficients.

We consider level set percolation for the Gaussian free field on the Euclidean lattice in dimensions larger than or equal to three. It had previously been shown by Bricmont, Lebowitz, and Maes that the critical level is non-negative in any dimension and finite in dimension three. Rodriguez and Sznitman have extended this result by proving that it is finite in all dimensions, and positive in all large enough dimensions.
We show that the critical parameter is positive in any dimension larger than or equal to three. In particular, this entails the percolation of sign clusters of the Gaussian free field.
This talk is based on joint work with A. Prévost (Köln) and P.-F. Rodriguez (Los Angeles).

Percolation on finite graphs is known to exhibit a phase transition similar to the Erdős-Rényi Random Graph in presence of sufficiently weak geometry. We focus on the Hamming graph H(d,n) (the cartesian product of d complete graphs on n vertices each) when d is fixed and n→∞. We identify the critical point p_c(d) at which such phase transition happens and we analyse the structure of the largest connected components at criticality. We prove that the scaling limit of component sizes is identical to the one for critical Erdős-Rényi components, while the number of surplus edges is much higher. These results are obtained coupling percolation to the trace of branching random walks on the Hamming graph.
Based on joint work with Remco van der Hofstad, Frank den Hollander and Tim Hulshof.

We consider the totally asymmetric simple exclusion process (TASEP) with a non-random initial condition that has a discontinuity (shock) in the particle density. If one inserts a "second class particle" in the system, it will follow the shock. For large time t, we show that the position of the second class particle fluctuates on the t^1/3 scale and we determine its limiting law. Joint work with Patrik Ferrari and Promit Ghosal.

Branching branching random walk and Brownian motion have been the subject of intensive research recently. We consider branching random walk and investigate the effect of introducing a spatially random branching environment. We are primarily interested in the position of the maximum particle, for which we prove a CLT. Our result correspond, on an analytic level, to a CLT for the front of the solutions to a randomized Fisher-KPP equation, and also to a CLT for the parabolic Anderson model.

A method for compression of large graphs and matrices to a block structure is proposed. Szemerédi's regularity lemma is used as a generic motivation of the significance of stochastic block models. Another ingredient of the method is Rissanen's minimum description length principle (MDL). We propose practical algorithms and provide theoretical results on the accuracy and consistency of the method.

We consider the problem of finding particular patterns in a realisation of a two-sided standard Brownian motion. Examples include two-sided Skorohod imbedding, the Brownian bridge and several other patterns, also in planar Brownian motion. The key tool here are recent allocation results in Palm theory.