# Bas Lodewijks

**Structural properties of explosive CMJ branching processes**

*22nd January 2024*

CMJ branching processes are a large class of continuous-time branching processes that have received a wealth of attention over time, in particular in the so called 'Malthusian regime'; branching processes that grow exponentially fast in time. In this talk I'd like to discuss some recent work on the 'explosive regime'; branching processes that grow infinitely large in finite time with positive probability. We study a large family of processes for which we obtain results on the structure of the branching process, stopped at the time it reaches an infinite size; in particular how the process has grown infinitely large. We then apply these results to super-linear preferenatial attachment trees with fitness, where we recover and generalise earlier work of Spencer and Oliveira.

Joint work with Tejas Iyer.

# Ed Crane

**Lumpings and Couplings of Markov Chains**

*5th February 2024*

Let X_n be a discrete time homogeneous Markov chain with state space A, and let f be a function mapping A to another set B. Typically, the sequence f(X_n) is not a Markov chain. But in some interesting cases it is also homogeneous Markov chain, for certain choices of the distribution of X_0. In that case, we say that f is a weak lumping of X.

In this talk I will discuss the theory of lumpings and two new applications.

The first application is joint work with Ander Holroyd and Erin Russell. Suppose you are given two homogeneous Markov chains with finite state spaces. You want to couple them subject to some constraints on the allowed coupled states and transitions, and you want the coupled process to be a homogeneous Markov chain. Can you decide whether this is possible, and if it is, how can you find such a coupling? We illustrate with examples involving forest fires, random walks on graphs, and queueing networks.

The second application is joint work with Alvaro Gutierrez, Erin Russell, and Mark Wildon. Define an irreducible random walk X_n on a finite group G, by letting X_0 be uniform on G and then setting

X_{n+1} = X_n g_n,

where (g_n) is an i.i.d. sequence of elements of G, independent of X_0. We use representation theory to determine all the possible distributions of g_1 for which the sequence of left cosets (X_n H) is a homogeneous Markov chain. Such sequences arise naturally in problems about card shuffling and pop-o-matic dice rollers.

Based on joint work with Erin Russell, Alexander Holroyd, Mark Wildon and Alvaro Gutierrez.

# Oliver Kelsey Tough

**Selection principles for the N-BBM and the Fleming-Viot particle system**

*12th February 2024*

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 on the real line, 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 tends to infinity 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 (time permitting) some of the ideas introduced to prove them.

This is based on joint work with Julien Berestycki.

# Louigi Addario-Berry

**The top eigenvalue of random trees**

*19th February 2024*

Let T_n be a uniformly random tree with vertex set [n]={1,...,n}. Let Δ_n be the largest vertex degree in T_n and let λ_n be the largest eigenvalue of T_n. We show that |λ_n - √Δ_n| tends to 0 in probability as n tends to infinity. The key ingredients of our proof are (a) the trace method, (b) a rewiring lemma that allows us to "clean up" our tree without decreasing its top eigenvalue, and (c) some careful combinatorial arguments.

This is joint work with Roberto Imbuzeiro Oliveira and Gabor Lugosi.

# Eleanor Archer

**Scaling limits of random spanning trees**

*26th February 2024*

A spanning tree of a finite connected graph G is a connected subgraph of G that contains every vertex and contains no cycles. A well-known result of Aldous states that the scaling limit of a uniformly chosen spanning tree of the complete graph is the Brownian tree. In fact, this statement is more general: the Brownian tree is the scaling limit of uniform spanning trees for a large set of high-dimensional graphs. In this talk, we'll try to explain this universal phenomenon. Time permitting, we will also discuss the scaling limits of non-uniform random spanning trees. Based on joint works with Asaf Nachmias and Matan Shalev.

# Victor Rivero

**Recurrent extensions and stochastic differential equations**

*4th March 2024*

In the 70's Itô settled the excursion theory of Markov processes, which is nowadays a fundamental tool for analyzing path properties of Markov processes. In his theory, Itô also introduced a method for building Markov processes using the excursion data, or by gluing excursions together, the resulting process is known as the recurrent extension of a given process. Since Itô's pioneering work the method of recurrent extensions has been added to the toolbox for building processes, which of course includes the martingale problem and stochastic differential equations. The latter are among the most popular tools for building and describing stochastic processes, in particular in applied models as they allow to physically describe the infinitesimal variations of the studied phenomena. In this work we answer the following natural question. Assume X is a Markov process taking values in R that dies at the first time it hits a distinguished point of the state space, say 0, which happens in a finite time a.s., that X satisfies a stochastic differential equation, and finally that X admits a recurrent extension, say Z, is a processes that behaves like Z up to the first hitting time of 0, and for which 0 is a recurrent and regular state. If any, what is the SDE satisfied by Z? Our answer to this question allows us to describe the SDE satisfied by many Feller processes. We analyze various particular examples, as for instance the so-called Feller brownian motions and diffusions, which include their sticky and skewed versions, and real valued Levy processes.

# Maria Deijfen

**Percolation in geometric random intersection graphs**

*11th March 2024*

An intersection graph is constructed by assigning each vertex a subset of some auxiliary set and then connecting two vertices if their subsets intersect. The model type has been popular in network modeling to describe networks arising from bipartite structures, for instance individuals who are connected if they share a social group, communication units connected via cell towers and scientists related through joint papers. We study a spatial version of the model type where both the vertex set and the auxiliary set are represented by Poisson
processes on R^{d,} giving rise to a variation of the random connection model. Our results concern local quantities (e.g. the degree distribution) and percolation properties of the resulting graph.

# Gerónimo Uribe Bravo

**A limit theorem for local times and an application to branching processes**

*18th March 2024*

Consider a stochastic process which is regenerative at some state and which is approximated by a sequence of discrete time regenerative processes. We give conditions so that the naturally defined local time (which counts the number of visits to the regenerative state) of the approximating processes has a scaling limit. We discuss examples dealing with Lévy processes but mostly focus on a recent application to Bienaymé-Galton-Watson processes with immigration.

# Christoforos Panagiotis

**Quantitative sub-ballisticity of self-avoiding walk on the hexagonal lattice**

*25th March 2024*

In this talk, we will consider the self-avoiding walk on the hexagonal lattice, which is one of the few lattices whose connective constant can be computed explicitly. This was proved by Duminil-Copin and Smirnov in 2012 when they introduced the parafermionic observable. In this talk, we will use the observable to show that, with high probability, a self-avoiding walk of length n does not exit a ball of radius n/logn. This improves on an earlier result of Duminil-Copin and Hammond, who obtained a non-quantitative o(n) bound. Along the way, we show that at criticality, the partition function of bridges of height T decays polynomially fast to 0. Joint work with Dmitrii Krachun.

# Jason Schweinsberg

**Asymptotics for the site frequency spectrum associated with the genealogy of a birth and death process**

*27th March 2024* (Note: This seminar is on a Wednesday, in 1W 2.03)

Consider a birth-death process started from one individual in which each individual gives birth at rate λ and dies at rate μ, so that the population size grows at rate r = λ - μ. Lambert (2018) and Harris, Johnston, and Roberts (2020) came up with methods for constructing the exact genealogy of a sample of size n taken from this population at time T. We use the construction of Lambert (2018), which is based on the coalescent point process, to obtain asymptotic results for the site frequency spectrum associated with this sample. We also explain how to apply these results to obtain a confidence interval for the growth rate of an exponentially growing tumor. This is joint work with Kit Curtius, Brian Johnson, and Yubo Shuai.

# Romain Panis

**An alternative approach for the mean-field behaviour of weakly self-avoiding walks in dimensions d>4**

*15th April 2024*

The weakly self-avoiding walk (WSAW) model on Z^{d} is a finite paths model in which self-intersections of the path are penalized. It naturally interpolates between the simple random walk model (no penalization) and the (truly) self-avoiding walk model (full penalization).

In high dimensions, it is expected that this model adopts a mean-field behaviour which means that it behaves (in some sense) similarly as if it was defined on a Cayley tree (or Bethe lattice). Brydges and Spencer (1985) introduced a method called "lace expansion" to perform the analysis of the mean-field regime of the WSAW model in dimensions d>4.

In this talk, I will present a new perspective on the study of the mean-field regime of the WSAW model which does not rely on lace expansion, and which is surprisingly short and simple.

Joint work with Hugo Duminil-Copin.

# Antal Jarai

**Activated random walk on the complete graph**

*22nd April 2024*

Activated random walk (ARW) is a discrete particle system living on a finite or infinite graph. Particles can be either 'active' or 'sleeping'. Active particles perform independent continuous-time simple random walks. Whenever a particle is alone on a vertex, it changes to the sleeping state at rate λ (a parameter of the model) and stops moving. As soon as an active particle jumps to a vertex occupied by a sleeping particle the sleeping particle turns active again. The interest in the model comes from its self-organised critical behaviour, in that sleeping particles obey a critical density (dependent on λ). For the complete graph on N vertices and one sink vertex, we show that the number of sleeping particles in the stationary distribution is λ/(1+λ) N with a correction of order square root of N log N. We identify the asymptotic constant for the correction in terms of an excursion event for the Orstein-Uhlenbeck process and prove its correctness. (Joint work with Christian Moench and Lorenzo Taggi.)

# Laure Marêché

**The self-repelling random walk with directed edges**

*29th April 2024 - Note this seminar will be held online only*

In this talk, we will consider a non-Markovian random walk such that the probability the walk goes to a given location is smaller if, in the past, it has often crossed the edge between the initial position and the target. The most studied such models are those in which edges are undirected. However, in 2008, Tóth et Vető introduced such a self-repelling random walk with directed edges, whose properties are very different from those of models with undirected edges. Despite the interest of such a behavior, very few results were obtained about this model afterwards. We will present new limit theorems for this random walk.