Statistical Mechanics of Ferromagnetism; Spherical Model as High-Density Limit

It is shown that there is a class of graphs of the Ising model (or (Heisenberg model for $T>\mathrm{}{T}_{\mathrm{Curie}}$) which is comprised of cycle graphs plus some excluded volume effects which sum to the spherical model. The spherical model, suitably generalized for $T<\mathrm{}{T}_{\mathrm{Curie}}$, was conjectured in a previous work to be the high-density limit of the Ising model, correct to $\frac{1}{z}$, where $z$ is the number of spins in the range of the exchange potential (not restricted to nearest neighbor interactions). $z^{\frac{1}{3}}$ measures the range of the exchange potential. This is now proved by examining the omitted graphs. The error is shown to be $O\left(\frac{1}{z^2}\right)$.