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Survival and complete convergence for a branching annihilating random walk

Branching systems with competition are interacting particle systems which have gained popularity as models for the reproduction of a spatial population with limited environmental resources. We study a branching annihilating random walk (BARW) in which particles move on the lattice and evolve 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 feature means that the system is not monotone and therefore the usual comparison methods are not applicable. We show that the system survives via coupling arguments and comparison with oriented percolation, making use of carefully defined density profiles which expand in time and are reminiscent of discrete travelling wave solutions. In the second part of the talk I will explain how a refinement of this technique can be employed to show complete convergence for the BARW in certain parameter regimes. The talk in based on a joint work with Matthias Birkner (JGU Mainz), Jiří Černý (University of Basel), Nina Gantert (TU Munich) and Pascal Oswald (University of Basel/JGU Mainz).