# HT17 Probability Workshops

**HILARY TERM 2017**

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

**Week 2 - Monday 23rd January**

**Speaker: Adrián Gonzalez-Casanova, WIAS Berlin**

**Title: ** Fixation and duality in a Xi coalescent model with selection

**Abstract: ** In this talk we will introduce a generalisation of the Wright Fisher model, for a population with finite size and non-overlapping generations, allowing for several types of selection as well as

simultaneous multiple mergers. The construction provides an almost sure dual relation between its frequency process and its ancestral process. The latter can be interpreted as a discrete analogue to the celebrated

ancestral selection graph. We will also study a two type frequency process with general selection and general coalescent mechanism, and investigates in which cases the selective type goes to fixation with

probability one. (This is talk is based in a joint project with Dario Spano)

**Week 3 - Monday 30th January**

**Speaker: James Martin, Department of Statistics, University of Oxford**

**Title: ** Stable matchings in R^d and on the Poisson-weighted infinite tree

**Abstract:** I'll talk about the model of "stable matchings". I'll start by mentioning classical ideas of stable marriage and related algorithms. Then I'll discuss multi-type stable matching problems for the points of a Poisson process in R^d, and an approach to studying them involving the "Poisson-weighted infinite tree" (or PWIT). The PWIT has been used in many settings as a scaling limit for complete graphs with independent edge weights. I'll explain how it also arises naturally as a scaling limit of Poisson processes in high-dimensional Euclidean space.

(Joint work with Alexander Holroyd and Yuval Peres.)

**Week 4 - Monday 6th February**

**Speaker: Paul Chleboun, Department of Statistics, University of Oxford**

**Title:** Static length scales and glassy dynamics in spin plaquette models.

**Abstract: ** Plaquette models are finite range spin systems which evolve according to Glauber dynamics. They have recently attracted a lot of attention in the physics literature because they are expected to exhibit glassy behavior, despite the absence of any disorder (like spin glasses) or hard constraints in the dynamics (like kinetically constrained models). In particular, we will focus on the square plaquette model and triangular plaquette model, which are expected to have dynamical behaviour similar to the East model and Fredrickson-Andersen model respectively. Despite having a very natural definition, and interesting dyanmics, there are currently surprisingly few rigorous results for plaquette models. We will first discuss equilibrium results related to the stationary distribution of these models, and then briefly discuss some results and conjectures on the dynamical behaviour.

This is work in progress with A. Faggionato, F. Martinelli, and C. Toninelli

**Week 5 - Monday 13th February**

**Speaker:** **Ngoc Tran, Hausdorff Center for Mathematics, Bonn, Germany**

**Title:** Stochastic tropical geometry

**Abstract:** Stochastic tropical geometry is the study of linear functionals of random tropical varieties. It is an exciting new field at the interface of algebraic geometry, probability and combinatorics, with connections to many others, such as economics and control theory. In this talk, we discuss some results on the simplest possible cases: the number of zeroes of a random tropical polynomial, and scaling limits of Poisson tropical plane curves. Both cases result in recursive, random partitions of space, whose generalizations are interesting processes in their own right. We will sketch the main proof ideas and discuss open questions in the field of stochastic tropical geometry.

**Week 6 - Monday 20th February**

**Speaker:** **Henning Sulzbach, University of Birmingham
Title: ** Galton-Watson trees and Apollonian networks

**In the analysis of critical Galton-Watson trees conditional on their sizes, two complementary approaches have proved fruitful over the last decades. First, the so-called size-biased tree allows the study of local properties such as node degrees. Second, the global scaling limit, the Continuum Random tree, provides access to the study of average or extreme node depths. In this talk, I will discuss the approximation of the tree by subtrees with bounded node degrees. Here, the so-called "heavy subtrees" play an important role. An application will be given in terms of uniform Apollonian networks.**

**Abstract:**The talk is based on joint work with Luc Devroye (McGill, Montreal) and Cecilia Holmgren (Uppsala).

**Week 7 - Monday 27th February**

**Speaker:** **Steven Pagett, University of Bath**

**Title: ** A phase transition in excursions from infinity of the "fast" fragmentation-coalescence process**Abstract**: An important property of Kingman's coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as `coming down from infinity'. In this talk I’ll show how robust this property is when paired with an extreme form of fragmentation, where each block fragments at constant rate, λ>0, into its constituent elements. The result being that there exists a phase transition in λ, between regimes where this ‘fast’ fragmentation-coalescence process is able to come down from infinity or not.

(Joint work with Tim Rogers, Andreas Kyprianou and Jason Schweinsberg)

**Week 8 - Monday 6th March**

**Speaker:** **Peter Mörters, University of Bath**

**Title: ** Metastability of the contact process on evolving scale-free networks**Abstract**: We study the contact process on scale-free networks with stationary dynamics based on simultaneous updating of all connections of a vertex. Our focus is on two types of inhomogeneous networks given by their underlying connection probabilities, the product type and the preferential attachment type. For these types we identify four possible strategies how the infection can survive on the network. We show that there is fast extinction of the infection when neither of the strategies is successful, otherwise there is slow extinction and the most successful strategy determines the asymptotics of the metastable density as the infection rate goes to zero. We identify the domains in which these strategies dominate in terms of phase diagrams for the exponent describing the decay of the metastable density.

This is joint work with Emmanuel Jacob (ENS Lyon) and Amitai Linker (Universidad de Chile).

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