The number of distinct sites visited in a random walk on a lattice

Journal Article · Sun Sep 01 00:00:00 EDT 1963 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.1704049· OSTI ID:4637387

Vineyard, George H.

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 {radical}8n/{pi})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, bodycentered 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. (auth)

• Research Organization: Brookhaven National Lab. (BNL), Upton, NY (United States)
• Sponsoring Organization: US Atomic Energy Commission (AEC)
• DOE Contract Number: AT(30-2)-Gen-16
• NSA Number: NSA-17-036846
• OSTI ID: 4637387
• Report Number(s): BNL-6853; JMAPA
• Journal Information: Journal of Mathematical Physics, Vol. Vol: 4, Issue 9; Other Information: BNL-6853. Orig. Receipt Date: 31-DEC-63; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

PHYSICS
DIFFERENTIAL EQUATIONS
INTEGRALS
LATTICES
MATHEMATICS
MOTION
NUMERICALS
STATISTICS