# Probability Workshops MT16

**MICHAELMAS TERM 2016**

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 1 - Monday 10th October**

**Speaker: Adrien Kassel, Department of Mathematics, ETH Zurich**

**Title: ** Random walk paths and discrete Gaussian free fields: a geometrical take on isomorphism theorems

**Abstract:** It is known, since the work of Dynkin building on pioneering ideas of Symanzik, that Markov processes and Markovian random fields are deeply related. In the simplest discrete setup of graphs, on which we shall focus in this talk, this relation links random walk paths to the discrete Gaussian free field.

At the formal level, this relation stems from the fact that a same operator, the discrete Laplacian, generates two types of processes: (1) by means of its semi-group (this is the random walk), (2) by means of its Dirichlet form (this is the discrete Gaussian free field).

At a refined probabilistic level, it is known that the local time (i.e.

the time spent at each vertex) of random walk paths is related to the square of the discrete Gaussian free field. This is Dynkin's isomorphism, which admits further variations and extensions, including theorems of Eisenbaum, Le Jan, and Sznitman.

In this talk, I will consider more general functionals of a path than its local time. These encode more faithfully the actual geometry of the trajectory and I will explain how they are related to vectorial versions of the Gaussian free field. Time permitting, I will say a few words about the related topics of gauge theory. No prior knowledge will be assumed.

Joint work with Thierry Lévy (Univ. Paris 6).

**Week 2 - Monday 17th October**

**Speaker: Juan Carlos Pardo Millán, CIMAT, Guanajuato**

**Title: ** Branching processes with interactions

**Abstract:** In this talk, I will introduce a branching process where individual die and reproduces as in the Bienayme-Galton-Watson process but also consider diferent types of interactions between individuals. We are interested in the long term behaviour of such processes and in order to do so, we will study its unique moment dual which turns out to be a jump-diffusion which can be "interpreted" as the frecuency of a phenotype with selective disadvantage in a given population.

This talk is based on a joint work with Adrian Gonzalez-Casanova and Jose Luis Perez

**Week 3 - Monday 24th October**

**Speaker: Will Perkins, School of Mathematics, University of Birmingham **

**Title:** On occupancy fractions, phase transitions, and extremal combinatorics

**Abstract: ** The hard sphere model is perhaps the oldest and most basic model of a fluid or a gas in statistical physics: a large number of identical spheres are placed uniformly at random in a large box conditioned on the event that none of them overlap. Despite the model's long history, a fundamental mathematical question remains open: is there a disordered/ordered phase transition as the density of spheres increases? I will explain related challenges in discrete models, why known techniques fail, and a new potential line of attack via the "occupancy fraction" of the model. I will then discuss how the occupancy fraction (and generalizations to "soft" models like the Potts model) can be used to prove results in extremal combinatorics.

**Week 4 - Monday 31st October**

**Speaker:** Matt Roberts, Department of Mathematical Sciences, University of Bath

**Title:** Exceptional times of the critical Erdős-Rényi graph

**Abstract:** It is well known that the largest components in the critical Erdős-Rényi graph have size of order n^{2/3}. We introduce a dynamic Erdős-Rényi graph by rerandomising each edge at rate 1, and ask whether there exist times in [0,1] at which the largest component is significantly larger than n^{2/3}.

**Week 5 - Monday 7th November**

** Speaker: Colin McDiarmid, Department of Statistics, University of Oxford** Clique colouring random graphs

Title:

**Graph colouring occupies a central place in theory and applications, in combinatorics, computer science and OR. We shall discuss clique colouring a graph G; that is, colouring the vertices so that no maximal clique is monochromatic. The least possible number of colours is the clique colouring number chi_c(G).**

**Abstract:**Typically, upper bounds on chi_c for random graphs are algorithmic. A key idea for clique colouring certain sparse random graphs is to construct triangle-free colour sets greedily. We shall discuss random perfect graphs, binomial random graphs and random geometric graphs. This is joint work with Nikola Yolov, and with Dieter Mitsche and Pawel Pralat.

**Week 6 - Wednesday 16th November (12 noon - 1 pm)**

**Speaker:** **Jean Bertoin, Institut für Mathematik, Universität Zürich**

**Title: ** Local explosion in growth-fragmentation processes (partly based on a joint work with Robin Stephenson)**Abstract**: Growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner.

In the self-similar case, it is known that a simple Malthusian condition ensures that the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. We shall present here the converse: when this Malthusian condition is not verified, then the growth-fragmentation process explodes almost surely. Our proof involves using the additive martingale to bias the probability measure and obtain a spine decomposition of the process, as well as properties of self-similar Markov processes.

**Week 7 - Monday 21st November**

**Speaker:**Noah Forman, Department of Statistics, University of Oxford

**Title:**Stationary diffusions on a space of interval partitions

**Abstract**: We construct two diffusions on a space of partitions of the unit interval. These are stationary with the law of the complement of the zero sets of Brownian motion and Brownian bridge, respectively. Our construction is based on decorating the jumps of a spectrally positive L\'evy process with independent continuous excursions. The processes of ranked interval lengths of our partitions belong to a two parameter family of diffusions introduced by Ethier and Kurtz (1981) and Petrov (2009). These are continuum limits of up-down Markov chains on Chinese restaurant processes. Our construction works towards building a diffusion on the space of real trees whose existence has been conjectured by Aldous.