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
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