Speaker
Description
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 models a population living in demes under a very strong form of local competition. For certain ranges of the parameters of the model, we show that the system dies out almost surely, or on the other hand survives with positive probability. In an even more restricted parameter range, we strengthen the survival results to convergence to a non-trivial invariant measure upon survival, and we show that the population invades space with a linear spreading speed. 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
@article{Birkner2024,
title = {Survival and complete convergence for a branching annihilating random walk},
volume = {34},
ISSN = {1050-5164},
url = {http://dx.doi.org/10.1214/24-AAP2105},
DOI = {10.1214/24-aap2105},
number = {6},
journal = {The Annals of Applied Probability},
publisher = {Institute of Mathematical Statistics},
author = {Birkner, Matthias and Callegaro, Alice and Černý, Jiří and Gantert, Nina and Oswald, Pascal},
year = {2024},
month = dec
}
