The Number of Distinct Sites Visited in a Random Walk on a Lattice

A general formalism is developed from which the average number of distinct sites visited in n steps by a random walker on a lattice can be calculated. The asymptotic value of this number for large n is shown to be ${\text{(8n/π)}}^{\frac{1}{2}}$ for a one‐dimensional lattice and cn for lattices of three or more dimensions. The constant c is evaluated exactly, with the help of Watson's integrals, for the simple cubic, body‐centered cubic, and face‐centered cubic lattices. An analogy is drawn with an electrical network in which unit resistors replace all near‐neighbor bonds in a lattice, and the resistance of such a network on each of the three cubic lattices is evaluated.