The high-temperature expansions of the partition function $Z$ and susceptibility $\chi$ of the Ising model and the number of self-avoiding walks $c_n$ and polygons $p_n$ are obtained exactly up to the eleventh order (in "bonds" or "steps") for the general $d$-dimensional simple hypercubical lattices. Exact expansions of $\ln Z$ and $\chi$ in power of $\frac{1}{q}$ where $q=2d$, and $\frac{1}{\sigma }$ where $\sigma =2d-1\mathrm{}$, for $T>\mathrm{}{T}_0$ are derived up to the fifth order. The zero-order terms are the Bragg-Williams and Bethe approximations, respectively. The Ising critical point is found to have the expansion ${\theta }_c=\frac{k{T}_c}{2dJ}=1-q^{-1\mathrm{}}-1\mathrm{}\frac{1}{3}{q}^{-2\mathrm{}}-4\mathrm{}\frac{1}{3}{q}^{-3\mathrm{}}-21\mathrm{}\frac{34}{45}{q}^{-4\mathrm{}}-133\mathrm{}\frac{14}{15}{q}^{-5\mathrm{}}-\cdots ,$ while for self-avoiding walks $\mu =limit\; of{\left|c_n\right|}^{\frac{1}{n}}\text{as}n\to \infty =\sigma [1-{\sigma }^{-2\mathrm{}}-2\mathrm{}{\sigma }^{-3\mathrm{}}-11\mathrm{}{\sigma }^{-4\mathrm{}}-62\mathrm{}{\sigma }^{-5\mathrm{}}-\cdots ].\mathrm{}$

Numerical extrapolation yields accurate estimates for ${\theta }_c$ and $\mu$ when $d=2\mathrm{} \mathrm{to} 6$ and indicates that $\chi$ diverges as ${(T-T_c)}^{-[1+\delta \mathrm{}(d\left)\right]}$ where $\frac{3}{\delta \mathrm{}\left(d\right)\mathrm{}}\simeq 4, 12, 32\pm 1, 80\pm 2, 188\pm 12, \cdots (d=2, 3\cdots ),$ and that $c_n\approx A{n}^{\alpha }{\mu }^n$ $(n\to \infty )$ with $\frac{1}{\alpha \mathrm{}\left(d\right)\mathrm{}}\simeq 3, 6, 14\pm 0.3, 32\pm 1.5, 72\pm 7, \cdots .$

• Received 17 July 1963

DOI:https://doi.org/10.1103/PhysRev.133.A224