Probability Seminar, Statistical Laboratory

# Probability Seminar, Statistical Laboratory

#### Tuesdays 4.30-5.30 pm, MR 12. Organisers:Nathanaël Berestycki, and Mike Tehranchi. Webpage of the probability group

Tea beforehand at 4pm in the common room, and drinks afterwards at the Punter.

### Lent 2011

 Date Speaker Title Notes 18 January Laurent Saloff-Coste (Cornell) Time inhomogeneous Markov chains: merging and stability (abstract) 25 January Johannes Ruf (Columbia) Hedging under arbitrage (abstract) Special time: 2pm. 25 January Antal Jarai (Bath) Abelian sandpiles on infinite graphs (abstract) 26 January Mark Kelbert (Swansea) Continuity of mutual information and the entropy-power inequality (abstract) Additional seminar, unusual day and time. 1 February Daniel Ueltschi (Warwick) Probabilistic representation of quantum correlations (abstract) 8 February Nicolas Fournier (Marne-La-Vallée) Scaling limits of forest fire processes. (abstract) 15 February Remi Peyre (Lyon) A probabilistic approach to Carne's bound (abstract). 22 February Wendelin Werner (Univerité Paris-Sud and ENS) Self-interacting random walks and their asymptotic behavior (abstract). 1 March Yuri Suhov (Cambridge). New results on spectra of multi-particle Schrödinger operators with random potentials (abstract). 8 March Julien Berestycki (Paris VI) The branching Brownian motion seen from its tip (abstract). 15 March Hugo Dominil-Copin (Geneva) Critical temperature of the square lattice Potts model (abstract).

#### 18 January (L. Saloff-Coste.) Time inhomogeneous Markov chains: merging and stability. Abstract:

We consider the problem of obtaining quantitative results regarding the behavior of well-behaved time inhomogeneous Markov chains on finite state spaces. The problem is difficult (Joint work with J. Zuniga).

#### 25 January (A. Jarai). Abelian sandpiles on infinite graphs. Abstract:

The Abelian sandpile model was introduced in the physics literature as a toy model of "self-organized criticality". Originally, it is defined as a Markov chain taking place on particle configurations on a finite graph. It received a lot of attention, since several of its characteristics, for example spatial correlations, were observed to follow power laws, akin to critical systems in statistical physics. In this talk, I will give an overview of results about limits on certain infinite graphs, including the d-dimensional integer lattice. I will also consider a perturbation of the model that has exponential decay of correlations, with rate of convergence estimates as the perturbation parameter vanishes. (In large part this is based on joint works with S.R. Athreya, R. Lyons, F. Redig and E. Saada.

#### 26 January (M. Kelbert). Continuity of mutual information and the entropy-power inequality. Abstract:

Consider an additive channel of information transmission with a continuous noise. The talk will focus on sufficient conditions for continuity of the input-output mutual information for large and small signal-to-noise ratios. This result leads to Shannon's entropy-power inequality which is a far-reaching generalisation of Minkowski's inequality from the classical analysis.

#### 1 February (Daniel Ueltschi) Probabilistic representation of quantum correlations. Abstract:

We consider lattice models of quantum spins that can be represented by random loop models in one more dimensions. Such models have been introduced over the years by Ginibre, Conlon-Solovej, Toth, and Aizenman-Nachtergaele. Long-range correlations are given by loops of macroscopic lengths. We argue that their joint distribution is given by the Poisson-Dirichlet distribution for random partitions. I will review some results that support this conjecture, results that come from either rigorous statistical mechanics, or from probability theory. This is joint work with C. Goldschmidt and P. Windridge.

#### 8 February (Nicolas Fournier). Scaling limits of forest fire processes. Abstract:

We consider the forest fire process on the integers: on each integer site, seeds and matches fall at random, according to some stationary renewal processes. When a seed falls on a vacant site, a tree immediately grows. When a match falls on an occupied site, a fire starts and destroys immediately the corresponding occupied connected component. We are interested in the asymptotics of rare fires. We prove that, under space/time re-scaling, the process converges (as matches become rarer and rarer) to a limit forest fire process. According to the tail distribution of the law of the delay between two seeds (on a given site), there are 4 possible limit processes.

#### 15 February (Remy Peyre). A probabilistic approach to Carne's bound. Abstract:

I this talk I will speak about a 1985 result from Carne & Varopoulos: consider a Markov chain on a graph (i.e. whose transitions always follow the edges) which is reversible with stationary measure $\mu$; then, denoting by $p^t(x,y)$ the probability that a chain starting at $x$ is at $y$ at time $t$, one has the Gaussian bound $p^t(x,y) \leq \sqrt{e} \big( \mu(y)/\mu(x) \big)^{1/2} \exp \big( -d(x,y)^2 / 2t \big) ,$ where $d(x,y)$ is the graph distance between $x$ and $y$. My goal will be to explain this result by probabilistic arguments (which was not the case of the original proof), close to the forward/ backward martingale decomposition. This approach will lead to a generalization of Carne's bound to the case where the particle can occasionally make a jump not following an edge. I will also explain how one can improve Carne's bound by a spectral factor, by considering the chain "conditioned to be recurrent".

#### 22 February (Wendelin Werner). Self-interacting random walks and their asymptotic behavior. Abstract:

I will discuss some joint work with Anna Erschler and Balint Toth, describing certain one-dimensional self-interacting walks on the set of integers, that choose to jump to the right or to the left randomly but influenced by the number of times they have previously jumped along the edges in the finite neighborhood of their current position. After describing the motivation to study such objects (the quest for new natural, continuous and local evolutions of functions that are neither deterministic PDEs nor stochastic PDEs), we will survey a variety of possible asymptotic behaviors (including some where the walks is eventually confined in an interval of arbitrarily large length) and the corresponding phase transitions.

#### 1 March (Y. Suhov). New results on spectra of multi-particle Schrödinger operators with random potentials. Abstract:

Random Schrödinger operators are popular in solid-state quantum physics because they provide accurate mathematical models for transitional phenomena occurring at the border between ordered and disordered systems (localisation {\it vs} delocalisation of eigenstates). Mathematically, the problem is to analyse properties of the spectrum of such an operator: this presents a challenge since new ideas and methods are required, combining Probability, Functional Analysis and -- occasionally -- other mathematical disciplines.

In this talk, I'll discuss new results on multi-particle random Schr\"odinger operators in a Euclidean space (joint works with several colleagues completed in 2010). The main result is that when the randomness is `strong', the spectrum near its lower edge is pure point with probability one and the corresponding eigenfunctions are exponentially localised.

No preliminary knowledge of Quantum Mechanics will be assumed.

#### 8 March (Julien Berestycki). The branching Brownian motion seen from its tip. Abstract:

It has been conjectured at least since a work of Lalley and Sellke (1987) that the branching Brownian motion seen from its tip (e.g. from its rightmost particle) converges to an invariant point process. The main goal of this talk is to present a proof of this fact which also gives a complete description of the limit object. The structure of this extremal point process turns out to be a certain Poisson point process with exponential intensity in which each atom has been decorated by an independent copy of an auxiliary point process. Joint work with Eric Brunet, Elie Aidekon and Zhan Shi.