# Probability Workshops

**MICHAELMAS TERM 2017**

Probability Workshops are held on **Mondays from 12:00 - 1:15 pm** during term time (unless indicated otherwise) in L5 at The Mathematical Institute, Andrew Wiles Building, Radcliffe Observatory Quarter.

**Week 1 - Monday 9th October**

**Speaker:** Christina Goldschmidt, Department of Statistics, Oxford

**Title:** Voronoi cells in the Brownian continuum random tree

**Abstract:** Take a uniform random tree with n vertices and select k of those vertices independently and uniformly at random; call them sites. (We assume that n is large and k is fixed, so that with high probability the sites are distinct.) Find the associated Voronoi cells: for each vertex in the tree, assign it to the cell of the site (or sites) which is closest in the graph distance. Now consider the vector of the proportions of the vertices lying in each of the k cells. We prove that this vector converges in distribution to the Dirichlet(1,1,…,1) distribution (that is, it is asymptotically uniform on the (k-1)-dimensional simplex). In fact, this is most easily formulated as a result about the scaling limit of the uniform random tree, namely the Brownian continuum random tree: if we pick k independent sites from the mass measure of the tree, their Voronoi cells have masses which are jointly Dirichlet(1,1,…,1) distributed. An analogue of this result also holds for (the scaling limit of) uniform unicellular random maps on surfaces of arbitrary genus. Joint work (in progress) with Louigi Addario-Berry, Omer Angel, Guillaume Chapuy and Éric Fusy.

**Week 2 - Monday 16th October**

**Speaker:** Andi Wang, Department of Statistics, Oxford

**Title:** Quasistationary Monte Carlo Methods

**Abstract:** Given a Markov process with an almost surely finite lifetime, what can be said about its law at some large time T, conditional on survival? This is the kind of question which the theory of quasistationarity aims to address. In my talk I will introduce this topic, and then focus on its recent application to statistics: the development of efficient Monte Carlo algorithms to sample from a given density \pi. I will first discuss some theoretical questions - when can we construct a killed diffusion with \pi as its quasistationary distribution? And what about rates of convergence? I will then move on to discuss how one might simulate from quasistationary distributions in practice.

Joint work with David Steinsaltz (Oxford), Martin Kolb (Paderborn) and Gareth Roberts (Warwick).

**Week 3 - Monday 23rd October**

**Speaker:** Jon Warren, Department of Statistics, Warwick

**Title:** KPZ and total positivity

Week 4 - Monday 30th October

**Speaker:** Costanza Benassi, Department of Mathematics, Warwick

**Title:** Random loop models: conjectures and results

**Abstract: ** Random loop models appear in a great variety of situations in both the probability and mathematical physics literature. Recently some striking conjectures have been suggested about the appearance of extended loops and the distribution of their lengths, which is expected to be a member of the Poisson Dirichlet distribution family. We propose a general class of models for interacting loops on a lattice, and we investigate how to validate these conjecture exploiting their relationship with a generalised random current model.

**Week 5 - Monday 6th November**

**Speaker:** Gesine Reinert, Department of Statistics, Oxford

**Title:** Stein's method and exhangeable pairs

**Abstract:**

**Week 6 - Monday 13th November**

**Speaker:** Andreas Kyprianou, Bath

**Title:** Terrorists never congregate in even numbers

**Abstract:** We analyse a class of fragmentation-coalescence processes defined on finite systems of particles organised into clusters. Coalescent events merge multiple clusters simultaneously to form a single larger cluster, while fragmentation breaks up a cluster into a collection of singletons. Under mild conditions on the coalescence rates, we show that the distribution of cluster sizes becomes non-random in the large-scale limit. Moreover, we discover that, in the limit of small fragmentation rate, these processes exhibit a universal heavy tailed distribution with exponent 3/2. In addition, we observe a strange phenomenon that if coalescence of clusters always involves 3 or more blocks, then the large-scale limit has no even sided blocks. Some complementary results are also presented for exchangeable fragmentation-coalescence processes on partitions of natural numbers. In this case one may work directly with the infinite system and we ask whether the process can come down from infinity. The answer reveals a remarkable dichotomy. This is based on two different pieces of work with Tim Rogers, Steven Pagett and Jason Schweinsberg.

**Week 7 - Monday 20th November**

**Speaker:** Nic Freeman, Sheffield

**Week 8 - Monday 27th November**

**Speaker:** Eleonora Kreacic, Department of Statistics, Oxford

Previous Workshops: TT17; HT17; MT16; TT16; HT16; MT15; TT15; HT15; MT14; TT14; HT14; MT13