Asymptotic form for random walk survival probabilities on three-dimensional lattices with traps

Abstract

The problem of calculating statistics of timeto-trapping of a random walker on a trapfilled lattice is of interest in solid state physics. Several authors have suggested approximate methods for calculating the average survival probabilities. Here, an exact asymptotic form for the probability that an n step random walk visits S. distinct sites is used to ascertain the validity of a simple approximation suggested by Rosenstock. For trap concentrations below 0.05, the relative error in using Rosenstock's approximation is less than 10%.There has been considerable recent interest in the problem of random walks on lattices with randomly distributed trapping sites (1-6). The original motivation for studying this problem was to model the trapping of mobile defects in crystals with point sinks (1, 2, 7), but many further applications have been found for this model. Rosenstock (2) studied the trapping model in analyzing kinetic experiments on luminescent organic materials, and Montroll (8, 9) considered the same model in connection with the kinetics of the conversion of light energy to oxygen in photosynthesis. Klafter and Silbey (6) give a good set of references to various applications of the random walk trapping model. Shuler et al. (10) have also dealt with the relationship between the expected number of distinct sites visited in a random walk and the mean time to trapping, in the context of the Montroll (8, 9) model in which there is one trapping site per unit cell. The present treatment is more general in the sense that it starts from the exact asymptotic distribution of the number of distinct sites visited in an n-step random walk rather than from its expected value.If the random walk takes place on a regular lattice and the probability that a given site is a trapping point is equal to c, then the probability that trapping occurs after step n is (1 -c)Sn where S. is the number of distinct sites visited during the course of an n-step random walk. Hence the expected time to trapping is (n ) = E (1-CM J=1

Search and Learning in the Immune System: Models of Immune Surveillance and Negative Selection

“…This approximation, called the Rosenstock approximation [157], is known to give good estimates for c < 0.05. In our case, c would denote the probability for each DC to activate the crawling T cell; given that perhaps 150 DCs are contacted per hour [11], and that the cell searches some hours before it is activated, c < 0.05 is a reasonable assumption in our case.…”