Detailed Calculations of the Number of Distinct Sites Visited in Random Walk on Several Two-Dimensional Substrate Lattices

Abstract

Calculations of the number of distinct sites visited in random walk on six physically different substrate lattices are carried out with exact, Monte Carlo and approximate analytical techniques. The fraction, λ, of the total possible sites, m, which are distinct is tabulated and plotted for all useful m for four lattice geometries and for 10≤m≤1000 for two more. The validity of this single valued characterization is examined and found to be very good, especially for larger m. Plots of parameters for functional application of the results are given.