IS13: Recent developments in branching structures

date: 7/17/2025, time: 16:00-17:30, room: ICS 140

Organizer: Sarah Penington (University of Bath)

Chair: Zsófia Talyigás (University of Vienna)

The Brownian Spatial Coalescent

Peter Koepernik (University of Oxford)

In this talk, I introduce a Markov coalescent process on the continuous \(d\)-dimensional torus called the Brownian spatial \(\Xi\)-coalescent. It describes the genealogies of the \(\Xi\)-Fleming-Viot process, which is the large population size limit of a class of spatial Cannings models in which individuals follow independent Brownian motions.

The Brownian spatial \(\Xi\)-coalescent can also be axiomatically characterised by a spatial notion of sampling consistency among a certain class of Markov coalescent processes, similarly to how the well-known \(\Xi\)-coalescent can be characterised through sampling consistency within the set of all non-spatial coalescents. An interesting consequence of this characterisation is that all spatial population models in which individuals follow independent Brownian motions have non-Markovian genealogies if the branching mechanism depends non-trivially on the spatial distribution, for example through local regulation.

Bibliography

\([1]\) Peter Koepernik. (2024). The Brownian Spatial Coalescent. arXiv:2401.08557.

Sharp LlogL condition for supercritical Galton-Watson processes with countable types

Mathilde André (ENS & University of Vienna)

We investigate Kesten–Stigum-like results for multi-type Galton–Watson processes with a countable number of types in a general setting, allowing us in particular to consider processes with an infinite total population at each generation. Specifically, a sharp \(L\log L\) condition is found under the only assumption that the mean reproduction matrix is positive recurrent in the sense of \([1]\). The type distribution is shown to always converge in probability in the recurrent case, and under conditions covering many cases it is shown to converge almost surely.

This is a joint work with Jean-Jil Duchamps (Université de Franche-Comté).

Bibliography

\([1]\) Vere-Jones, D. (1967) Ergodic properties of nonnegative matrices. I. Pacific Journal of Mathematics, 22(2):361–386.

The longest increasing subsequence of Brownian separable permutons

William Da Silva (University of Vienna)

The Brownian separable permutons form a one-parameter family of permutons indexed by \(p\in(0,1)\) and connected to trees, which are the universal scaling limits of pattern-avoiding permutations. In this talk, we will be interested in the length of the longest increasing subsequence (LIS) of a permutation \(\sigma_n\) of size \(n\) sampled from the Brownian permuton of parameter \(p\). We give an answer to the celebrated Ulam-Hammersley problem in this context: what is the typical behaviour of \(\text{LIS}(\sigma_n)\) as \(n\) goes to infinity? In fact, we prove a scaling limit result for the LIS:

Theorem. We have the almost sure convergence \[\frac{\mathrm{LIS}(\sigma_n)}{n^{\alpha}} \underset{n\rightarrow \infty}{\longrightarrow} X,\] where \(\alpha=\alpha(p)\) is the unique solution in the interval \((1/2,1)\) to the equation \[\frac{1}{4^{\frac{1}{2\alpha}}\sqrt{\pi}}\,\frac{\Gamma\big(\tfrac{1}{2}-\tfrac{1}{2\alpha}\big)}{\Gamma\big(1-\tfrac{1}{2\alpha}\big)}=\frac{p}{p-1},\] and \(X=X(p)\) is a non-deterministic and a.s. positive and finite random variable, which is a measurable function of the Brownian separable permuton.

A significant portion of the talk will be dedicated to our motivation behind the problem, emphasising connections to various objects in probability, such as random decorated trees, random graphs, directed planar maps and SLE/LQG. The talk is based on a recent joint work with Arka Adhikari, Jacopo Borga, Thomas Budzinski and Delphin Sénizergues \([1]\).

Bibliography

\([1]\) A. Adhikari, J. Borga, T. Budzinski, W. Da Silva and D. Sénizergues. The longest increasing subsequence of Brownian separable permutons. arxiv:2506.19123