# Hilary 2016

**HILARY 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 - Thursday 21st January**

**Speaker: ** Anita Behme, Technische Universität München

**Title: ** On the mapping associated to exponential functionals of Levy processes

**Abstract:** For a given bivariate L\'evy process $(\xi,\eta)$ the random variable $\int_0^\infty e^{-\xi_{s-}} d\eta_s$, whenever it exists, is called the exponential functional $(\xi,\eta)$. Its distribution appears in various applications, e.g. as the stationary solution of a generalized Ornstein–Uhlenbeck process.

In this talk, we shall be interested in properties of the mapping $\Phi_\xi$, which associates to every L\'evy process $\eta$ the distribution of the corresponding exponential functional for a fixed $\xi$, independent of $\eta$. While the case of $\xi_t = t$ is well studied and gives rise to all self-decomposable distributions, much less is known for general $\xi$.

**Week 2 - Monday 25th January**

**Speaker: ** Bati Sengul, University of Bath

**Title:** Cutoff for conjugacy-invariant random walks on the permutation group

**Abstract: ** Suppose we take a large deck of cards and consider a random walk where at each step, it swaps two uniformly chosen cards. How long does it take before the deck of cards looks like the uniform distribution? Diaconis and Shahshahani showed that the answer is (1/2) n log n. This left open possible generalisations, for example the random walk where at each step we pick k cards uniformly and swap their positions (the first card goes into the second cards position etc.). I will present the recent work which solves this problem for a general class of random walks which solves a conjecture raised by the work of Diaconis and Shahshahani. This is joint work with Nathanael Berestycki.

**Week 3 - Monday 1st February**

**Speaker: ** Alexander Watson, University of Manchester

**Title:** Growth-fragmentation models, random and deterministic

**Abstract: ** We look at models of fragmentation with growth. In such a model, one has a number of independent cells, each of which grows continuously in time until a fragmentation event occurs, at which point the cell splits into two child cells of a smaller mass. Each of the children is independent and behaves in the same way as its parent. The rate of fragmentation may be infinite, and fragmentation may be homogeneous (where the rate does not depend on the mass of the cell) or self-similar (where the rate is a power of the mass). This is a random model; looking at its mean-field behaviour gives the growth-fragmentation equation, which is a deterministic PDE. Variants of the equation have been studied recently from an analytic perspective, but we use a probabilistic approach, akin to that taken by Haas for the pure fragmentation equation. We describe probabilistic solutions to the equation, using the compensated fragmentation processes recently described by Bertoin. One interesting phenomenon is that, in some self-similar cases, we see spontaneous generation of positive solutions from zero initial mass. Based on joint work with Jean Bertoin (University of Zurich).

**Week 4 - Monday 8th February**

**Speaker:** Martin Balasz, University of Bristol

**Title:** How to initialise a second class particle?

**Abstract:** This talk will be on interacting particle systems. One of the best known models in the field is the simple exclusion process where every site has 0 or 1 particles. It has long been established that under certain rescaling procedure this process converges to solutions of a deterministic nonlinear PDE (Burger's equation). Particular types of solutions, called rarefaction fans, arise from decreasing step initial data.

Second class particles are probabilistic objects that come from coupling two interacting particle systems. They are very useful and their behaviour is highly nontrivial.

The beautiful paper of P. A. Ferrari and C. Kipnis connects the above: they proved that the second class particle of simple exclusion chooses a uniform random velocity when started in a rarefaction fan. The extremely elegant proof is based, among other ideas, on the fact that increasing the mean of a Bernoulli distribution can be done by adding or not adding 1 to the random variable.

For a long time simple exclusion was the only model with an established large scale behaviour of the second class particle in its rarefaction fan. I will explain how this is done in the Ferrari-Kipnis paper, then show how to do this for other models that allow more than one particles per site. The main issue is that most families of distributions are not as nice as Bernoulli in terms of increasing their parameter by just adding or not adding 1. To overcome this we use a signed, non-probabilistic coupling measure that nevertheless points out a canonical initial probability distribution for the second class particle. We can then use this initial distribution to greatly generalize the Ferrari-Kipnis argument. I will conclude with an example where the second class particle velocity has a mixed discrete and continuous distribution. Joint work with Attila László Nagy

**Week 5 - Monday 15th February**

**Speaker:** Ben Hambly, Mathematical Institute, University of Oxford

**Title:** Spectral asymptotics for somerandom fractals

**Abstract: ** A classical theorem of Weyl's shows that informationabout the geometry of a bounded domain can be obtained from the asymptotics of the eigenvalue counting function for the Laplace operator on the domain. We will consider the behaviour of the second order term in the asymptotics in the case of some simple domains with random fractal boundary and for the continuum random tree. By developing results for general branching processes we can obtain conditions under which there are central limit theorems for the spectral asymptotics and we give examples for which there are and are not CLTs.

**Week 6 - Monday 22nd February**

**Speaker:** Igor Kortchemski, Ecole Polytechnique, Université Paris-Saclay

**Title:** Self-similar scaling limits of Markov chains on the positive integers

**Abstract:** We will be interested in the scaling limits of Markov chains on the set of positive integers for which, loosely speaking, large jumps are rare and occur at a rate that behaves like a negative power of the current state, and such that small positive and negative steps of the chain roughly compensate each other. We identify three different regimes (loosely speaking the transient, the recurrent and the positive-recurrent regimes) in which the scaling limit exhibits different behavior. This has for instance applications to the study of Markov chains with asymptotically zero drifts such as Bessel-type random walks and to the structure of large random planar maps. This is based on joint works with Jean Bertoin and Nicolas Curien.

**Week 7 - Monday 29th February**

**Speaker:** Stephen Muirhead, Mathematical Institute, University of Oxford

**Title:** Quenched localisation in slowly varying trap models

**Abstract:** Trap models with slowly varying trap distributions constitute one of the three basic regimes of trap models (along with integrable and stable traps), and arise naturally in the study of certain random walks in random media, such as biased random walks on critical structures, spin-glass dynamics on sub-exponential time scales, and recurrent random walks in random environments. In this talk I will discuss recent progress (joint with David Croydon) on understanding quenched localisation properties of slowly varying trap models, which turn out to be surprisingly delicate. Our main result concerns a simple effective trap model -- the Bouchaud trap model on the positive integers -- for which we demonstrate that there exist slowly varying trap distributions such that quenched localisation occurs on exactly N sites, for any integer N greater or equal to two. A key ingredient is an observation about the almost sure limit superior of the sum/max ratio of i.i.d. sequences of slowly varying random variables, which appears to be new.

**Week 8 - Monday 7th March**

**Speaker:**Tom Hutchcroft, University of British Columbia

**Title:**Update Tolerance in Uniform Spanning Forests

**Abstract:**The uniform spanning forests (USFs) of an infinite graph G are defined to be infinite volume limits of uniformly chosen spanning trees of finite subgraphs of G. These limits can be taken with respect to two extremal boundary conditions, yielding the free uniform spanning forest (FUSF) and wired uniform spanning forest (WUSF). While the wired uniform spanning forest has been quite well understood since the seminal paper of Benjamini, Lyons, Peres and Schramm (’01), the FUSF is less understood, and some very basic questions about it remain open. In this talk I will introduce a new tool in the study of USFs, called update tolerance, and describe how update tolerance can be used to prove, among other things, that the FUSF has either one or infinitely many connected components on any infinite Cayley graph, and that components of either the FUSF and WUSF are indistinguishable from each other by invariantly defined properties on any infinite Cayley graph. Another crucial component of these proofs is the Mass-Transport Principle, which I will also give an introduction to.

Based in part on joint work with Asaf Nachmias.