IS19: Branching and Interacting Particle Systems

date: 7/14/2025, time: 14:00-15:30, room: ICS 13

Organizer: Dariusz Buraczewski & Piotr Dyszewski (University of Wrocław)

Chair: Piotr Dyszewski (University of Wrocław)

Discounted tree sums in branching random walks

YUEYUN HU (Université Paris XIII)

This talk is based on a joint work with Elie Aïdékon and Zhan Shi. Let \((V(u),\, u\in {\mathcal T})\) be a (supercritical) branching random walk and \((\eta_u,\,u\in {\mathcal T})\) be marks on the vertices of the tree, distributed in an i.i.d. fashion. Following Aldous and Bandyopadhyay (2005), for each infinite ray \(\xi\) of the tree, we associate the discounted tree sum \(D(\xi)\) which is the sum of the \(e^{-V(u)}\eta_u\) taken along the ray. The paper deals with the finiteness of \(\sup_\xi D(\xi)\). To this end, we study the extreme behaviour of the local time processes of the paths \((V(u),\,u\in \xi)\). It answers a question of Nicolas Curien, and partially solves Open Problem 31 of Aldous and Bandyopadhyay (2005). We also present several open questions.

A branching annihilating random walk model

Alice Callegaro (Technical University of Munich)

We study a branching-annihilating random walk in which particles evolve on the lattice in discrete generations. Each particle produces a Poissonian number of offspring which independently move to a uniformly chosen site within a fixed distance from their parent’s position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. This can be thought of as a very strong form of local competition and implies that the system is not monotone. For certain ranges of the parameters of the model, the system dies out almost surely or survives with positive probability. In an even more restricted parameter range, survival can be strengthened to complete convergence with a non-trivial invariant measure and in dimension one, to a shape theorem. A central tool in the proof is comparison with oriented percolation on a coarse-grained level, using carefully tuned density profiles which expand in time and are reminiscent of discrete travelling wave solutions.

Bibliography

\([1]\) Matthias Birkner, Alice Callegaro, Jiří Černý, Nina Gantert, and Pascal Oswald. "Survival and complete convergence for a branching annihilating random walk." Ann. Appl. Probab., 34(6), 2024, 5737-5768.

Explosion of Crump-Mode-Jagers processes with critical immediate offspring

Matthias Meiners (University of Gießen)

In my talk, I will consider the phenomenon of explosion in general (Crump-Mode-Jagers) branching processes, which refers to the event where an infinite number of individuals are born in finite time. In a critical setting where the expected number of immediate offspring per individual is exactly 1, whether or not explosion occurs depends on the fine properties of the reproduction point process. I will review some known results and explain recent results in this area. In particular, I present a necessary and sufficient condition in the case where the reproduction point process is Poissonian.