Article,
Random Walks on Lattices. II
Journal of Mathematical Physics, 6 (2): 167--181 (Dec 22, 2004)
DOI: 10.1063/1.1704269
Abstract
Formulas are obtained for the mean first passage times (as well as their dispersion) in random walks from the origin to an arbitrary lattice point on a periodic space lattice with periodic boundary conditions. Generally this time is proportional to the number of lattice points. The number of distinct points visited after n steps on a k‐dimensional lattice (with k ≥ 3) when n is large is a 1 n + a 2 n ½ + a 3 + a 4 n −½ + …. The constants a 1 − a 4 have been obtained for walks on a simple cubic lattice when k = 3 and a 1 and a 2 are given for simple and face‐centered cubic lattices. Formulas have also been obtained for the number of points visited r times in n steps as well as the average number of times a given point has been visited. The probability F(c) that a walker on a one‐dimensional lattice returns to his starting point before being trapped on a lattice of trap concentration c is F(c) = 1 + c/(1 − c) log c. Most of the results in this paper have been derived by the method of Green's functions.
