Ising-model, spin-spin correlations on the hypercubical lattices

The first seven terms of the high-temperature series expansion for the true range of correlation are derived for general dimension on the family of hypercubical lattices. Analysis of this series shows that a cross-over phenomenon occurs at high dimension so that for $d=\infty$ one obtains for the correlation-length critical index $\nu =1\mathrm{}$ instead of the classical value $\nu =\frac{1}{2}$. This behavior is also illustrated by a more tractable, analogous, random-walk problem. The behavior of the critical temperature as a function of dimension is discussed. All present evidence is consistent with an essential singularity at $d=4\mathrm{}$.