## Probability in the North East workshop

#### 24 June 2024

Organizers: Sunil Chhita and Andrew Wade (Durham).

**Venue:** Department of Mathematical Sciences, Durham University.
Talks in MCS2068.

Registration is free, but please do **register** by sending an email to the organizers at pine.admin@durham.ac.uk) by **Friday 14th June**.

## Programme

10:00–10:50

In joint work with Zemer Kosloff, we will discuss the dynamical properties of a seemingly innocuous perturbation of a sequence of independent and identically distributed (iid) coin-flips to one that is no longer stationary. In the stationary case, Ornstein proved that iid systems are completely classified up to isomorphism by their Shannon entropy. We will find that in the nonstationary case, the usual entropy theory no longer applies, but we will recover an explicit version of the Sinai factor theorem that allows us to generate iid randomness from a nonstationary source.

11:20–11:50

Interfaces in many planar statistical mechanics models at criticality can be described by an SLE$_{\kappa}$ curve, for some value of $\kappa$. In 2009, Makarov and Smirnov asked if the interfaces remain when the models are near criticality and if so, whether these interfaces can be described in an explicit way. In this talk, I will consider the case of the massive Gaussian free field (GFF), which is a near-critical perturbation of the Gaussian free field. I will construct a coupling between a massive GFF and a random curve in which the curve can be seen as an interface of the field. I will then show that in this coupling, the curve can be described explicitly: it has the law of massive SLE$_4$, which is the unique law on non-self-intersecting curves for which a certain observable is a martingale.

12:00–12:50

Branching processes naturally arise as pertinent models in a variety of situations such as cell division, population dynamics and nuclear fission. For a wide class of branching processes, it is common that their first moment exhibits a Perron Frobenius-type decomposition. That is, the first order asymptotic behaviour is described by a triple $(\lambda, \varphi, \eta)$, where $\lambda$ is the leading eigenvalue of the system and $\varphi$ and $\eta$ are the corresponding right eigenfunction and left eigenmeasure respectively. Thus, obtaining good estimates of these quantities is imperative for understanding the long-time behaviour of these processes. In this talk, we discuss various Monte Carlo methods for estimating this triple.

This talk is based on joint work with Alex Cox (University of Bath) and Denis Villemonais (UniversitĂ© de Lorraine).

14:00–14:50

The selection problem is to show, for a given branching particle system with selection, that the stationary distribution for a large but finite number of particles corresponds to the travelling wave of the associated PDE with minimal wave speed. This had been an open problem for any such particle system.

The $N$-branching Brownian motion with selection ($N$-BBM) is a particle system consisting of $N$ independent particles that diffuse as Brownian motions in $\mathbb{R}$, branch at rate one, and whose size is kept constant by removing the leftmost particle at each branching event. We establish the following selection principle: as $N\rightarrow\infty$ the stationary empirical measure of the $N$-particle system converges to the minimal travelling wave of the associated free boundary PDE. Moreover we will establish a similar selection principle for the related Fleming-Viot particle system with drift $-1$, a selection problem which had arisen in a different context.

We will discuss these selection principles, their backgrounds, and some of the ideas introduced to prove them.

This is based on joint work with Julien Berestycki.

15:00–15:30

In this talk, we study the limiting behavior of the perimeter and diameter functionals of the convex hull spanned by the first $n$ steps of two planar random walks. As the main results, we obtain the strong law of large numbers and the central limit theorem for the perimeter and diameter of these random sets.

15:40–16:40

First we study the simplest possible self-similar fragmentation process. The process starts at time zero with a single fragment of size $1$, which has an $\exp(1)$ lifetime before splitting into two fragments of size $\frac{1}{2}$. Thereafter, for a parameter $q<1$, a fragment of size $2^{-n}$ has an $\exp(q^n)$ lifetime before splitting into two fragments of sizes $2^{-n-1}$. We find that at large times the sizes of the largest and smallest fragment in the system can be characterised with high probability to specific integer powers of $\frac{1}{2}$. Our approach draws on connections with branching random walks, point processes and $q$-combinatorics.

We close the talk by discussing some recent work on infinite activity fragmentation processes, where all fragments in the system continuously (but randomly) crumble in time.

This is joint work with Piotr Dyszewski, Nina Gantert, Sandra Palau, Joscha Prochno and Dominik Schmid.