Excluded-Volume Problem and the Ising Model of Ferromagnetism

The relationship between the excluded-volume problem for a discrete random walk on a lattice and the corresponding Ising model of ferromagnetism is investigated. Systematic methods are presented for the construction of rigorous lower bounds to the limit μ=lim_n→∞(cₙ₊₁cₙ), where cₙ is the number of n-step self-avoiding walks on a given lattice. In this way Temperley's conjecture that μ=coth(JkTC), where TC is the Curie temperature of the corresponding Ising-model ferromagnet, is disproved. The series cₙ for various two- and three-dimensional lattices have been enumerated exactly for values of n from ten to twenty. Extrapolation of these series, by procedures known to be valid from exact Ising-model results, yields more accurate values of μ than Wall's statistical calculations and also shows that cₙ~n^αμⁿ where α1/3 for plane lattices and α1/7 for three-dimensional lattices. This means that the entropy of the nth "link" of a polymer molecule in solution should vary as δSₙ=klnμ+kαn. The relevance of these results to the interpretation of the boundary tension of the Ising model, to the critical behavior of gases, and to the mean square size of a polymer molecule is discussed briefly.