Invariance relations for random walks on hexagonal lattices

Abstract

Abstract We consider the problem of random walks on finite, N =(2 k ×2 k ) hexagonal lattices with a single, deep trap, and subject to periodic boundary conditions. An exact expression is obtained for calculating the invariance relation linking the set M of n th nearest-neighbor sites surrounding the trapping site, viz., (2M−3)N−{2M+6+3M[ ln (M/6)/ ln (2)]}. This result may be used to obtain approximate values of the overall mean walklength 〈 n 〉. The results are compared with exact numerical results, with the predictions of the asymptotic expression of Montroll and Weiss, and linked to current studies in nanotube chemistry.