A sandpile on a graph is an integer-valued function on the vertices. It evolves according to local moves called topplings. Some sandpiles stabilize after a finite number of topplings, while others topple forever. For any sandpile s_0 if we repeatedly add a grain of sand at an independent random vertex, we eventually reach a "threshold state'' s_T that topples forever. Poghosyan, Poghosyan, Priezzhev and Ruelle conjectured a precise value for the expected amount of sand in s_T in the limit as s_0 tends to negative infinity. I will outline how this conjecture was proved in http://arxiv.org/abs/1402.3283 by means of a Markov renewal theorem. This talk will be elementary and all sandpile terms will be defined.

The simple harmonic urn is a two-colour urn model whose sample paths approximate the phase portrait of the simple harmonic oscillator. We start with N red balls and 0 blue balls. Each time a red ball is picked from the urn, it is returned along with a blue ball. Each time a blue ball is picked from the urn, it is returned and a red ball is removed from the urn. We stop when the last red ball is removed. By coupling the process to the renewal process with uniform inter-arrival times, we obtain precise estimates on the mean and variance of the number of blue balls remaining, and an exact description of its distribution in terms of Eulerian numbers. Whenever the urn becomes monochromatic, we can continue the process by exchanging the roles of the two colours. Is the resulting process recurrent or transient? Come to the talk to find out!

I will prove relations between the k-way discrepancy md_k(C) of a rectangular array C of nonnegative entries and its spectra. In one direction, irrespective of the size of C, I give the following estimate for the kth largest non-trivial singular value of the normalized table: s_k <= 9 md_k(C) (k + 2 - 9k ln md_k(C)), provided 0 < md_k(C) < 1 and k <= rank(C). The other direction and Butler's result for the k=1 case will also be considered. The result naturally extends to the singular values of the normalized adjacency matrix of a weighted undirected or directed graph.

We combine the loss-network dynamics introduced in [1] with the framework of Pirogov-Sinai theory to obtain new proofs of the stability results given by this theory, which replace the traditional use of cluster expansions with tools from the theory of stochastic processes. This dynamical approach allows us to enlarge the standard range of stability in some models while also yielding a perfect simulation algorithm for the infinite-volume stable phases of systems in the non-uniqueness regime.
[1] Fernández, Roberto, Ferrari, Pablo A., Garcia, Nancy L. Loss network representation of Peierls contours. Ann. Probab. 29 (2001), no. 2, 902-937.

We examine two different models with long memory. The true self-avoiding walk (TSAW) is a class of self-repelling random walks, for which we prove central limit approximation using Kipnis-Varadhan technology. For a class of density-dependent population generalized semi-Markov processes (PGSMP, a class of population models where non-Markovian transitions are also allowed), we derive and prove the mean-field limit using probability concentration results.

The Kac-Ward formula expresses the partition function of the Ising model on a finite planar graph as the square root of the determinant of a certain operator acting locally on the edges of the graph. I will first present a new short and simple proof of this classical result. Second, I will explain how this framework can be generalized and conveniently reinterpreted in the context of the double-Ising model, the spin model obtained by taking the pointwise product of +/- spins in two independent Ising models. In this model, the interfaces separating + and - form a collection of random loops, which have been conjectured by David Wilson to have a precise universal scaling limit. I will present Wilson's conjecture and discuss some relevant facts in the light of the above framework. Joint work in progress with Dmitry Chelkak (ETH-ITS Zurich & Steklov Institute St.Petersburg) and David Cimasoni (University of Geneva).

The Lorentz mirror model can be thought of as a percolation problem or as a deterministic dynamical system in a quenched random environment. We show, incorporating results obtained from numerical simulations, that the density of open paths in any finite box tends to 0 as the box size tends to infinity, for any mirror probability.
Joint work with David Sanders, available at http://arxiv.org/pdf/1406.4796v1.pdf [arxiv.org]

We consider the Erdos-Renyi graph G(n,p), conditioned to contain no cycles. We use results of Britikov and Luczak to show an Aldous-type limit for the depth-first search within the same critical window as in the classical case, to a limiting process with Brownian excursions.