## Abstract

We consider the survival of a random walker in a finite lattice with periodic boundary conditions. The initial position of the random walker is uniformly distributed on the lattice with respect to the trap. We show that the survival of a random walker, 〈U_{n}>, can be exactly related to the expected number of distinct sites visted on a trap-free lattice by 〈U_{n}〉=1-〈S_{n}〉/N^{D} (*) where N^{D} is the number of lattice points in D dimensions. We then analyze the behavior of 〈S_{n}〉 in any number of dimensions by using Tauberian methods. We find that at sufficiently long times 〈S_{n}〉 decays exponentially with n in all numbers of dimensions. In D = 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid for N^{2}≫N ≫ 1 when D = 1 and N ln N ≫n≫ 1 when D = 2. No such crossover exists when Z≥3. The form of (*) suggests that the single trap approximation is indeed a valid low-concentration limit for survival on an infinite lattice with a finite concentration of traps.

Original language | English |
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Pages (from-to) | 191-199 |

Number of pages | 9 |

Journal | Journal of Statistical Physics |

Volume | 40 |

Issue number | 1-2 |

DOIs | |

State | Published - Jul 1985 |

## Keywords

- Random walks
- Tauberian theorems
- trapping