This series was previously called Probability Workshops and is now called Probability Seminars. You can find a list of past (from Trinity Term 2018) and upcoming Probability Seminars on Oxford Talks. The Probability Seminars take place in the Mathematical Institute.
Hilary Term 2018
Speaker: Eleonora Kreacic, University of Oxford
Title: The spread of fire on a random multigraph
Abstract: We study a model for the destruction of a random network by fire. Suppose that we are given a multigraph of minimum degree at least 2 having real-valued edge-lengths. We pick a uniform point from along the length and set it alight; the edges of the multigraph burn at speed 1. If the fire reaches a vertex of degree 2, the fire gets directly passed on to the neighbouring edge; a vertex of degree at least 3, however, passes the fire either to all of its neighbours or none, each with probability 1/2. If the fire goes out before the whole network is burnt, we again set fire to a uniform point. We are interested in the number of fires which must be set in order to burn the whole network, and the number of points which are burnt from two different directions. We analyse these quantities for a random multigraph having n vertices of degree 3 and alpha(n) vertices of degree 4, where alpha(n)=o(n), with i.i.d. standard exponential edge-lengths. Depending on whether alpha(n) >> \sqrt{n} or alpha(n)=O(\sqrt{n}), we prove that as n tends to infinity these quantities converge jointly in distribution when suitably rescaled to either a pair of constants or to (complicated) functionals of Brownian motion. This is joint work with Christina Goldschmidt.
Speaker: Minmin Wang, University of Bath
Title: Scaling limits of critical inhomogeneous random graphs
Abstract: Branching processes are known to be useful tools in studying random graphs, for instance in understanding the phase transition phenomenon in the asymptotics sizes of their connected components. In this talk, I’d like to discuss some applications of Galton—Watson trees (genealogy of branching process) in studying the geometrical aspects of random graphs. In particular, we will look at the so-called Poisson random graph, a random graph model which generalises the Erdos-Renyi graph G(n, p) and which has been previously studied by Aldous and Limic for its close connection with the multiplicative coalescents. Relying upon an embedding of the graph into a Galton-Watson forest, we can identify the scaling limits of these graphs inside the critical window.
Based on a joint work with Nicolas Broutin and Thomas Duquesne.
Speaker: Sandra Palau, University of Bath
Title: Extinction properties and asymptotic behaviour of multi-type continuous state branching processes
Abstract: In this talk we are going to study multi-type continuous state branching processes. Under mild conditions, we will see that there exists a lead eigenvalue associated with the first-moment semigroup. The sign of this eigenvalue distinguishes between the cases where there is extinction and exponential growth. In the supercritical case, we will give the a.s. rate of growth and the convergence proportion of each type.
Speaker: Varun Kanade, Department of Computer Science & LMH, University of Oxford
Title: On coalescence time in graphs--When is coalescing as fast as meeting?
Abstract: Coalescing random walks is a fundamental stochastic process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite recent progress such as by Cooper et al. and Berenbrink et al., the coalescence time for graphs such as binary trees, d-dimensional tori, hypercubes and more generally, vertex-transitive graphs, remains unresolved. We provide a powerful toolkit that results in tight bounds for various topologies including the aforementioned ones.
The meeting time is defined as the worst-case expected time required for two random walks to arrive at the same node at the same time. As a general result, we establish that for graphs whose meeting time is only marginally larger than the mixing time (a factor of log^2n), the coalescence time of n random walks equals the meeting time up to constant factors. This upper bound is complemented by the construction of a graph family demonstrating that this result is the best possible up to constant factors. For almost-regular graphs, we bound the coalescence time by the hitting time, resolving the discrete-time variant of a conjecture by Aldous for this class of graphs. Finally, we prove that for any graph the coalescence time is bounded by O(n^3). By duality, our results give bounds on the voter model. Based on joint work with Frederik Mallmann-Trenn (MIT) and Thomas Sauerwald (Cambridge).
Speaker: Robin Stephenson, University of Oxford
Title: Convergence of bivariate Markov chains to multi-type self-similar processes, and scaling limits of recursively growing trees.
Abstract: We investigate the scaling limits of sequences of random trees obtained by recursive constructions. One crucial element is identifying their so-called Markov branching structure, which can be seen as a generalised Galton-Watson-type property, where individuals have mass and spread their mass to their children. The general study of Markov branching trees involves specific bivariate Markov chains, of which we then show the convergence to multi-type self-similar processes under various conditions.
Speaker: Aleksander Klimek, Mathematical Institute, University of Oxford
Title: Selection in fluctuating environment
Abstract: We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment. We consider a population with no spatial structure, modelled by an adaptation of the Lambda (or generalised) Fleming-Viot process, and derive a stochastic differential equation as a scaling limit. This amounts to a limit result for a Lambda-Fleming-Viot process in a rapidly fluctuating random environment. We then extend to a population that is distributed across a spatial continuum, which we model through a modification of the spatial Lambda-Fleming-Viot process with selection. In this setting we show that the scaling limit is a stochastic partial differential equation. In dimensions greater than one, the ‘genetic drift’ disappears in the scaling limit, but here we retain some stochasticity due to the fluctuations in the environment, resulting in a stochastic PDE driven by a noise that is white in time but coloured in space. We discuss a possible extensions of this work, including description of the dynamics of a rare type in those models. Joint with Niloy Biswas, Alison Etheridge and Jonathan Chetwynd-Diggle.
Speaker: Matt Junge, Duke University
Title: Parking
Abstract: A car or parking spot is placed at each vertex of an infinite graph. Cars drive in search of unparked spots. When a car finds one, both the car and spot are removed. Three distinct traffic flow phases arise as we vary the initial ratio of cars to spots. There is particularly intriguing behavior at criticality. Joint work with Michael Damron, Janko Gravner, Hanbeck Lyu, and David Sivakoff: arXiv id: 1710.10529.
Michaelmas Term 2017
Speaker: Gesine Reinert, Department of Statistics, Oxford
Title: Stein's method and exhangeable pairs
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.
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.
Trinity Term 2017
Speaker: Amaury Lambert, Laboratoire de Probabilités & Modèles Aléatoires, UPMC Univ Paris 06
Title: Random ultrametric trees and applications
Abstract: Ultrametric trees are trees whose leaves lie at the same distance from the root. They are used to model the genealogy of a population of particles co-existing at the same point in time. We show how the boundary of an ultrametric tree, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. We display a variety of examples of random combs and explain how they can be used in applications. In particular, we review some old and more recent results regarding the genetic structure of the population when throwing neutral mutations on the skeleton of the tree.
Speaker: Raphael Forien, CMAP - École Polytechnique, France
Title: Gene flow across geographical barriers
Abstract: Barriers to gene flow are physical or biological obstacles which locally reduce migration and genetic exchanges between different regions of a species' habitat. Genetic relatedness between individuals can be deduced from the distribution of the positions of their ancestors back in time. In the presence of obstacles to migration, these ancestral lineages can be modelled by simple random walks outside of a bounded region around the obstacle. We present a continuous real-valued process which is obtained as a scaling limit of these random walks, and we give several constructions of this process.
Speaker: Mikolaj Kasprzak, Department of Statistics, University of Oxford
Title: Diffusion approximations via Stein's method and time changes
Abstract: We extend the ideas of (Barbour, 1990) and use Stein's method to obtain a bound on the distance between a scaled time-changed random walk and a time-changed Brownian Motion. We then apply this result to bound the distance between a time-changed compensated scaled Poisson process and a time-changed Brownian Motion. This allows us to bound the distance between the Moran model with mutation and Wright-Fisher diffusion with mutation upon noting that the former may be expressed as a difference of two time-changed Poisson processes and the diffusive part of the latter may be expressed as a time-changed Brownian Motion. The method is applicable to a much wider class of examples satisfying the Stroock-Varadhan theory of diffusion approximation.
Speaker: Matthias Winkel, Department of Statistics, University of Oxford
Title: Gromov-Hausdorff-Prokhorov convergence of vertex cut-trees of n-leaf Galton-Watson trees
Abstract: In this paper we study the vertex cut-trees of Galton-Watson trees conditioned to have n leaves. This notion is a slight variation of Dieuleveut's vertex cut-tree of Galton-Watson trees conditioned to have n vertices. Our main result is a joint Gromov-Hausdorff-Prokhorov convergence in the finite variance case of the Galton-Watson tree and its vertex cut-tree to Bertoin and Miermont's joint distribution of the Brownian CRT and its cut-tree. The methods also apply to the infinite variance case, but the problem to strengthen Dieuleveut's and Bertoin and Miermont's Gromov-Prokhorov convergence to Gromov-Hausdorff-Prokhorov remains open for their models conditioned to have n vertices. This is joint work with Hui He, Beijing Normal University.
Speaker: Jeffrey Rosenthal, University of Toronto
Title: Conditions for Convergence of Adaptive MCMC Algorithms
Abstract: Markov chain Monte Carlo (MCMC) algorithms, such as the Metropolis Algorithm and the Gibbs Sampler, are an extremely popular method of approximately sampling from complicated probability distributions. Adaptive MCMC attempts to automatically modify the algorithm while it runs, to improve its performance on the fly. However, such modifications can destroy the convergence properties necessary for the algorithm to be valid. In this talk, we first illustrate MCMC algorithms using simple graphical Java applets. We then discuss adaptive MCMC, and present examples and theorems concerning its convergence and efficiency. We close with some recent results which provide more easily verifiable sufficient conditions for convergence.