# Probability Workshops MT15

**MICHAELMAS TERM 2015**

Probability Workshops are held on **Mondays from 12:00 - 1:15 pm** during term time in L4 at The Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter.

**Week 1 - Monday 12th October**

**Speaker: ** Agelos Georgakopoulos, University of Warwick

**Title: ** Group Walk Random Graphs

**Abstract:** I will discuss a construction of finite 'geometric' random graphs motivated by the study of random walks on infinite groups. This construction has connections to other topics, including the Poisson boundary and Sznitman's random interlacements (which I will try to introduce in a gentle way) http://arxiv.org/abs/1506.02697.

**Week 2 - Monday 19th October**

**Speaker: ** Nikos Zygouras, University of Warwick

**Title:** From disorder relevance to the 2d Stochastic Heat Equation

**Abstract: ** We consider statistical mechanics models defined on a lattice, in which disorder acts as an external random field. Such models are called disorder relevant, if arbitrarily weak disorder changes the qualitative properties of the model. Via a Lindeberg principle for multilinear polynomials we show that disorder relevance manifests itself through the existence of a disordered high-temperature limit for the partition function, which is given in terms of Wiener chaos and is model specific.

When disorder becomes marginally relevant a fundamentally new structure emerges, which leads to a universal scaling limit for all different (currently directed polymer) models that fall in this class. A notable such representative is the two dimensional SHE with multipicative space-time white noise (which in the SPDE language is characterised as "critical"). In this case certain analogies with Gaussian Multiplicative Chaos and log-correlated Gaussian fields appear.

Based on joint works with Francesco Caravenna and Rongfeng Sun.

**Week 3 - Monday 26th October**

**Speaker: ** Alex Hening, Department of Statistics, University of Oxford

**Title:** The free path in a high velocity random flight process associated to a Lorentz gas in an external field

**Abstract: ** We investigate the asymptotic behavior of the free path of avariable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight.

**Week 4 - Monday 2nd November**

**Speaker:** Paul Chleboun, University of Warwick

**Title:** Relaxation and mixing of kinetically constrained models.

**Abstract: ** We study the relaxation and out-of-equilibrium dynamics of a family of kinetically constrained models (KCMs) called the d-dimensional East-like processes. KCMs are spin systems on integer lattices, where each vertex is labelled either 0 or 1, which evolve according to a very simple rule: i) with rate one and independently for each vertex, a new value 1/0 is proposed with probability 1-q and q respectively; ii) the proposed value is accepted if and only if the neighbouring spins satisfy a certain constraint. Despite of their apparent simplicity, KCMs pose very challenging and interesting problems due to the hardness of the constraints and lack of monotonicity. The out-of-equilibrium dynamics are extremely rich and display many of the key features of the dynamics of real glasses, such as; an ergodicity breaking transition at some critical value, huge relaxation times close to the critical point, and dynamic heterogeneity (non-trivial spatio-temporal fluctuations of the local relaxation to equilibrium). We discuss recent advances on the out-of-equilibrium dynamics of the East-like processes, including the dependence of the relaxation and mixing time on the system size, density, and dimension. We also look at simulations which motivate some interesting limit shape conjectures.

This is joint work with Alessandra Faggionato and Fabio Martinelli.

**Week 5 - Monday 9th November**

**Speaker:** Tom Kurtz, University of Wisconsin - Madison

**Title:** Stochastic equations for processes built from bounded generators

**Abstract:** The generator for a pure jump process with bounded jump rate is a bounded operator on the space of measurable functions. For any such process, it is simple to write a stochastic equation driven by a Poisson random measure. Uniqueness for both the stochastic equation and the corresponding martingale problem is immediate, and consequently, the martingale problem and the stochastic equation are equivalent in the sense that they uniquely characterize the same process. A variety of Markov processes, including many interacting particle models, have generators which are at least formally given by infinite sums of bounded generators. In considerable generality, we can write stochastic equations that are equivalent to these generators in the sense that every solution of the stochastic equation is a solution of the martingale problem and every solution of the martingale problem determines a weak solution of the stochastic equation.

**Week 6 - Monday 16th November**

**Speaker:** Sarah Penington, Department of Statistics

**Title:** Curvature Flow, Allen-Cahn and the Spatial Lambda Fleming Viot process

**Abstract:** Hybrid zones are interfaces between populations which occur when two species interbreed, but the hybrids have a lower evolutionary fitness. We can model this situation using the spatial Lambda Fleming Viot process (SLFV), and study the limiting behaviour using a dual process of branching and coalescing random walks.

We use a duality relation with a Branching Brownian motion to give a probabilistic proof of a PDE result (originally proved by Chen) that in solutions to an Allen-Cahn equation, an interface forms which moves approximately according to curvature flow.

Our proof of Chen's result is flexible enough that we can also apply it to the SLFV dual to prove that the hybrid zone evolves approximately according to curvature flow.

Joint work with Alison Etheridge and Nic Freeman.

**Week 7 - Monday 23rd November**

**Speaker:** Vittoria Silvestri, Faculty of Mathematics, University of Cambridge

**Title:** Hastings-Levitov growth and the GFF

**Abstract:** I will discuss one instance of the so called Hastings-Levitov planar aggregation model, consisting of growing random clusters on the complex plane, built by iterated composition of random conformal maps.

In 2012 Norris and Turner proved that in the case of i.i.d. maps the limiting shape of these clusters is a disc. In this talk I will show that the fluctuations around this asymptotic behaviour are given by a random holomorphic Gaussian field F on {|z|>1}, of which I will provide an explicit construction. The boundary values of F perform a Gaussian Markov process on the space of distributions, which is conveniently described as the solution of a stochastic PDE. When the cluster is allowed to grow indefinitely, this process converges to the restriction of the whole plane Gaussian Free Field to the unit circle.

**Week 8 - Monday 30th November**

**Speaker:** Olly Johnson, University of Bristol

**Title:** Algorithms and bounds for the group testing problem

**Abstract:** The group testing problem was introduced by Dorfman in the 1940s, and gives a model for isolating a small number of infected members of a larger population. I will review recent work on this problem, and explain some new algorithms which can be proved to perform well in certain sparsity regimes. To complement this, I will explain how a channel coding argument of Polyanskiy, Poor and Verdu gives an upper bound on the success rate that can be achieved by any algorithm, by a comparison with a certain statistical hypothesis test.

**PLEASE NOTE THERE WILL BE AN ADDITIONAL WORKSHOP IN WEEK 9. DETAILS GIVEN BELOW.**

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**Week 9 - Thursday 10th December**

**Speaker:** Harry Crane, Department of Statistics and Biostatistics, Rutgers, The State University of New Jersey

**Title:** Partial symmetries in random structures

**Abstract:** The study of exchangeability emerges from classical considerations of symmetry and the principle of indifference in inductive inference. In practice, many statistical and scientific problems exhibit only partial symmetry determined by some underlying structure in a population. As a simple example, consider measurements X_1, X_2, ... taken on a population of men and Y_1, Y_2, ... taken on a population of women. Without further information, we may assume the distribution of (X_1,X_2,...; Y_1,Y_2,...) is symmetric under independent relabeling of the X's and Y's, but not under arbitrary relabeling of the entire sequence. In general applications, the symmetries may be more complex, leading to the notion of `relative exchangeability', a type of partial exchangeability which reflects distributional invariance with respect to the symmetries of another structure. I will discuss recent work in this area, including generic representations for relatively exchangeable structures and applications to the study of certain combinatorial stochastic processes.

Previous Workshops: TT15; HT15; MT14; TT14; HT14; MT13