On the survival probability of a random walk in a finite lattice with a single trap |
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Authors: | George H Weiss Shlomo Havlin Armin Bunde |
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Institution: | (1) National Institutes of Health, 20205 Bethesda, Maryland;(2) Department of Physics, Bar-Ilan University, Ramat-Gan, Israel;(3) Center for Polymer Studies and Department of Physics, Boston University, 62215 Boston, Massachusetts;(4) Present address: Fakultät für Physik, Universität Konstanz, D-7750 Konstanz, Federal Republic of Germany |
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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 (*) whereN
D is the number of lattice points inD dimensions. We then analyze the behavior of Sn in any number of dimensions by using Tauberian methods. We find that at sufficiently long times S
n decays exponentially withn in all numbers of dimensions. InD = 1 and 2 dimensions there is an intermediate behavior which can be calculated and is valid forN
2 N 1 whenD = 1 andN lnN n 1 whenD = 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. |
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Keywords: | Random walks trapping Tauberian theorems |
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