2023/04/18 by Matthias Birkner, Birkner, Matthias, Alice Callegaro +7
Mathematics · Physics and Astronomy · #60K35 (Primary) #92D25 (Secondary) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2304.09127
openalex publication_date 2023/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a discrete-time branching annihilating random walk (BARW) on the d-dimensional lattice. Each particle produces a Poissonian number of offspring with mean μ which independently move to a uniformly chosen site within a fixed distance R from their parent's position. Whenever a site is occupied by at least two particles, all the particles at that site are annihilated. We prove that for any μ>1 the process survives when R is sufficiently large. For fixed R we show that the process dies out if μ is too small or too large. Furthermore, we exhibit an interval of μ-values for which the process survives and possesses a unique non-trivial ergodic equilibrium for R sufficiently large. We also prove complete convergence for that case.