Invariance relations for random walks on hexagonal lattices

We consider the problem of random walks on finite, N=(2k×2k) 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 nth 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.