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 =lim₍ (c₍+₁{c₍}), where c₍ is the number of n-step self-avoiding walks on a given lattice. In this way Temperley's conjecture that =coth (Jk{T₂}), where T₂ is the Curie temperature of the corresponding Ising-model ferromagnet, is disproved. The series c₍ 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 than Wall's statistical calculations and also shows that c₍n^^n where 13 for plane lattices and 17 for three-dimensional lattices. This means that the entropy of the nth "link" of a polymer molecule in solution should vary as S₍=kln+kn. 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.