Excluded-Volume Effect for Two- and Three-Dimensional Lattice Models

Abstract

An analysis is undertaken of mean‐square sizes of self‐avoiding walks enumerated exactly on various two‐ and three‐dimensional lattices. It is suggested that for all three‐dimensional lattices the mean‐square size tends to an asymptotic relation 〈rn 2〉≃A 1 n 6/5+A 2, while for two‐dimensional lattices the corresponding relation is 〈rn 2〉≃B 1 n 3/2+B 2 n. The constants A 1, B 1 decrease as the coordination of the lattice increases. Estimates of A 1, A 2 are given for the fcc, bcc, simple cubic, and diamond lattices, and of B 1, B 2 for the triangular and simple quadratic lattices.