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.

### Michaelmas 2009

 Date Speaker Title Notes 13 October. Stas Volkov (Bristol). The simple harmonic urn (abstract) 20 October Neil O'Connell (Warwick) Directed polymers and the quantum Toda lattice (abstract) 27 October Anne Fey (Eindhoven) Sandpiles, Staircases and Self-organized criticality (abstract) 3 November Alison Etheridge (Oxford) Drift, draft and structure: modelling evolution in a spatial continuum (abstract). 10 November Mike Tehranchi (Cambridge) Put-call symmetry (abstract). 17 November Aleksander Mijatovic (Imperial College) On the martingale property of certain local martingales (abstract). 24 November Yuri Suhov (Cambridge) Concentration inequalities for random matrix and operator eigenvalues (abstract). 27 November Aad van der Vaart (Amsterdam) Click here for details. Joint with statistics, note special day. 2 December Augusto Teixeira (ETH) Percolation of random interlacements under small intensities (abstract) Note special day, and time: 4pm. In Newton Institute Gatehouse. 4 December Oren Louidor (NYU) Directed polymers in random environment with heavy tails. (abstract) Note special day, and time: 2.30pm. In Newton Institute Gatehouse.

#### Stas Volkov. The simple harmonic urn (October 13). Abstract:

The simple harmonic urn is a discrete-time stochastic process approximating the phase portrait of the simple harmonic oscillator. This urn is a version of a two-colour generalized Polya urn with negative-positive reinforcements, and in a sense it can be viewed as a "marriage" between the Friedman urn and the OK Corral model, where we restart the process each time it hits a border by switching the colours of the balls. We show the transience of the process using various couplings with birth and death processes and renewal processes. It turns out that the simple harmonic urn is just barely transient, as a minor modification of the model makes it recurrent, and for a good reason, as the embedded urn process is, in fact, a Lamperti-type random walk. We also show links between this model and oriented percolation, as well as a few other interesting processes. This is joint work with E. Crane, N. Georgiou, R. Waters and A. Wade.

#### Neil O'Connell. Directed polymers and the quantum Toda lattice. Abstract:

We give a characterization of the law of the partition function of a Brownian directed polymer model in terms of the eigenfunctions of the quantum Toda lattice. The proof is via a multidimensional generalization of theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion.

#### Anne Fey. Sandpiles, Staircases and Self-organized criticality (October 27). Abstract:

In the fixed-energy sandpile model, `activity density' jumps are observed as the particle density increases, in some cases even seeming to form a devil's staircase. After presenting new results on this topic, we will focus on the first jump, from zero to nonzero activity density. The particle density where this transition takes place, plays a role in a popular heuristic argument to explain self-organized criticality in sandpile models: it is argued that another, differently defined particle density is in fact the same. This conjecture was supported by simulations and generally believed to be true. However, we can now for several cases exactly calculate both densities, giving very close, but unequal values. (joint work with Lionel Levine and David Wilson)

#### Alison Etheridge. Drift, draft and structure: modelling evolution in a spatial continuum (November 3rd). Abstract:

One of the outstanding successes of mathematical population genetics is Kingman's coalescent. This process provides a simple and elegant description of the genealogical trees relating individuals in a sample of neutral genes from a panmictic population, that is, one in which every individual is equally likely to mate with every other and all individuals experience the same conditions. But real populations are not like this. Spurred on by the recent flood of DNA sequence data, an enormous industry has developed that seeks to extend Kingman's coalescent to incorporate things like variable population size, natural selection and spatial and genetic structure. But a satisfactory approach to populations evolving in a spatial continuum has proved elusive. In this talk we describe the effects of some of these biologically important phenomena on the genealogical trees before describing a new approach (joint work with Nick Barton, IST Austria) to modelling the evolution of populations distributed in a spatial continuum.

#### Mike Tehranchi. Put-call symmetry (November 10th). Abstract:

The pricing formulae for put and call options in the Black--Scholes model satisfy a certain symmetry relationship. There has been growing interest in asset price models that exhibit this put-call symmetry since, in the context of such models, certain barrier options can be replicated by a semi-static trading strategy in the underlying stock. This talk will survey these results as well as recent results on characterizing models that exhibit put-call symmetry.

#### Aleksander Mijatovic. On the martingale property of certain local martingales (November 17th). Abstract:

The stochastic exponential Z of a continuous local martingale M is itself a continuous local martingale. In this talk we describe a necessary and sufficient condition for the process Z to be a true martingale in the case where M_t=int_0^t b(Y_u)dW_u and Y is a one-dimensional diffusion driven by a Brownian motion W. Furthermore, we give a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function $b$ and the drift and diffusion coefficients of Y. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting. This is joint work with Misha Urusov.

#### Yuri Suhov. Concentration inequalities for random matrix and operator eigenvalues. Abstract:

In the random matrix theory it is often important to assess the probability that an eigenvalue comes close to a given point or that two (or more) eigenvalues come close to each other. I will address this question in the context of recent progress in the theory of random Schroedinger operators. No preliminary knowledge of Quntum Mechanics will be assumed.

#### Augusto Teixeira. Percolation of random interlacements under small intensities. Abstract:

The model of random interlacements was recently introduced by Alain-Sol Sznitman as a natural tool to understand the trace left by a random walk in a discrete cylinder or in a discrete torus. In these contexts, this model describes the microscopic "texture in the bulk" left by the random walk when it is let run up to certain time scales. In this talk we are going to discuss some percolative properties of the vacant set of random interlacements under small intensities (e.g. the size of a finite vacant cluster). The results which will be presented could shed some light on problems such as how a random walk trajectory disconnects a discrete cylinder into two infinite connected components.

#### Oren Louidor. Directed polymers in random environment with heavy tails. Abstract:

We study the model of Directed Polymers in Random Environment in 1+1 dimensions, where the environment is i.i.d. with a site distribution having a tail that decays regularly polynomially with power -\alpha, where \alpha \in (0,2). After proper scaling of temperature \beta^{-1}, we show strong localization of the polymer to an optimal region in the environment where energy and entropy are best balanced. We prove that this region has a weak limit under linear scaling and identify the limiting distribution as an (\alpha, \beta)-indexed family of measures on Lipschitz curves lying inside the 45^{\circ}-rotated square with unit diagonal. In particular, this shows order of n for the transversal fluctuations of the polymer. If (and only if) \alpha is small enough, we find that there exists a random critical temperature above which the effect of the environment is not macroscopically noticeable. The results carry over to higher dimensions with minor modifications. Joint work with A. Auffinger .