ANALYTIC SOLUTION FOR RANDOM-WALKS ON FINITE D-DIMENSIONAL LATTICES WITH A SINGLE TRAP

We have developed a combined theoretical and numerical method for constructing analytic solutions to a class of stochastic (random-walk) problems set in lattices of arbitrary (integral) dimension. The method is illustrated for the case of a particle diffusing in a finite d-dimensional space defined by a hypercube with N = 3d lattice sites and subject to periodic boundary conditions. Using results generated for d less-than-or-equal-to 10, it is shown how analytic expressions can be obtained for the mean number <n> of steps before trapping at a centrosymmetric site in a hyperspace of high dimension, with explicit results given for d less-than-or-equal-to 5. The results obtained cast light on certain geometrical aspects of the (strictly ergodic) flow of a phase point in a high-dimensional space, on an invariance relation first proved by Montroll [J. Math. Phys. 6, 165 (1965)], and on some consequences of compactification.