Survival and complete convergence for a branching annihilating random walk
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866929325819822080 |
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| author | Birkner, Matthias Callegaro, Alice Černý, Jiří Gantert, Nina Oswald, Pascal |
| author_facet | Birkner, Matthias Callegaro, Alice Černý, Jiří Gantert, Nina Oswald, Pascal |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_09127 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Survival and complete convergence for a branching annihilating random walk Birkner, Matthias Callegaro, Alice Černý, Jiří Gantert, Nina Oswald, Pascal Probability 60K35 (Primary), 92D25 (Secondary) 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. |
| title | Survival and complete convergence for a branching annihilating random walk |
| topic | Probability 60K35 (Primary), 92D25 (Secondary) |
| url | https://arxiv.org/abs/2304.09127 |