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 $\ensuremath{\mu}={\mathrm{lim}}_{n\ensuremath{\rightarrow}\ensuremath{\infty}}(\frac{{c}_{n+1}}{{c}_{n}})$, where ${c}_{n}$ is the number of $n$-step self-avoiding walks on a given lattice. In this way Temperley's conjecture that $\ensuremath{\mu}=coth(\frac{J}{k{T}_{C}})$, where ${T}_{C}$ is the Curie temperature of the corresponding Ising-model ferromagnet, is disproved. The series ${c}_{n}$ 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 $\ensuremath{\mu}$ than Wall's statistical calculations and also shows that ${c}_{n}\ensuremath{\sim}{n}^{\ensuremath{\alpha}}{\ensuremath{\mu}}^{n}$ where $\ensuremath{\alpha}\ensuremath{\simeq}\frac{1}{3}$ for plane lattices and $\ensuremath{\alpha}\ensuremath{\simeq}\frac{1}{7}$ for three-dimensional lattices. This means that the entropy of the $n\mathrm{th}$ "link" of a polymer molecule in solution should vary as $\ensuremath{\delta}{S}_{n}=k\mathrm{ln}\ensuremath{\mu}+\frac{k\ensuremath{\alpha}}{n}$. 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.