### Abstract

Using a high performance computer cluster, we run simulations regarding an open problem about d-dimensional critical branching random walks in a random IID environment The environment is given by the rule that at every site independently, with probability p ∈ [0;1], there is a cookie, completely suppressing the branching of any particle located there. The simulations suggest self averaging: the asymptotic survival probability in n steps is the same in the annealed and the quenched case; it is 2/qn, where q: = 1 - p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probability q for every particle at every iteration.

Original language | English (US) |
---|---|

Pages (from-to) | 169-193 |

Number of pages | 25 |

Journal | Monte Carlo Methods and Applications |

Volume | 17 |

Issue number | 2 |

DOIs | |

State | Published - Jun 2011 |

### Fingerprint

### Keywords

- Branching random walk
- Catalytic branching
- Critical branching
- Mild obstacles
- Random environment
- Simulation

### ASJC Scopus subject areas

- Statistics and Probability
- Applied Mathematics

### Cite this

*Monte Carlo Methods and Applications*,

*17*(2), 169-193. https://doi.org/10.1515/MCMA.2011.008

**Critical branching random walk in an IID environment.** / Engländer, János; Sieben, Nandor.

Research output: Contribution to journal › Article

*Monte Carlo Methods and Applications*, vol. 17, no. 2, pp. 169-193. https://doi.org/10.1515/MCMA.2011.008

}

TY - JOUR

T1 - Critical branching random walk in an IID environment

AU - Engländer, János

AU - Sieben, Nandor

PY - 2011/6

Y1 - 2011/6

N2 - Using a high performance computer cluster, we run simulations regarding an open problem about d-dimensional critical branching random walks in a random IID environment The environment is given by the rule that at every site independently, with probability p ∈ [0;1], there is a cookie, completely suppressing the branching of any particle located there. The simulations suggest self averaging: the asymptotic survival probability in n steps is the same in the annealed and the quenched case; it is 2/qn, where q: = 1 - p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probability q for every particle at every iteration.

AB - Using a high performance computer cluster, we run simulations regarding an open problem about d-dimensional critical branching random walks in a random IID environment The environment is given by the rule that at every site independently, with probability p ∈ [0;1], there is a cookie, completely suppressing the branching of any particle located there. The simulations suggest self averaging: the asymptotic survival probability in n steps is the same in the annealed and the quenched case; it is 2/qn, where q: = 1 - p. This particular asymptotics indicates a non-trivial phenomenon: the tail of the survival probability (both in the annealed and the quenched case) is the same as in the case of non-spatial unit time critical branching, where the branching rule is modified: branching only takes place with probability q for every particle at every iteration.

KW - Branching random walk

KW - Catalytic branching

KW - Critical branching

KW - Mild obstacles

KW - Random environment

KW - Simulation

UR - http://www.scopus.com/inward/record.url?scp=84858647587&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858647587&partnerID=8YFLogxK

U2 - 10.1515/MCMA.2011.008

DO - 10.1515/MCMA.2011.008

M3 - Article

VL - 17

SP - 169

EP - 193

JO - Monte Carlo Methods and Applications

JF - Monte Carlo Methods and Applications

SN - 0929-9629

IS - 2

ER -