Ising Model and Self-Avoiding Walks on Hypercubical Lattices and "High-Density" Expansions

The high-temperature expansions of the partition function Z and susceptibility of the Ising model and the number of self-avoiding walks c₍ and polygons p₍ are obtained exactly up to the eleventh order (in "bonds" or "steps") for the general d-dimensional simple hypercubical lattices. Exact expansions of lnZ and in power of 1q where q=2d, and 1 where =2d-1, for T>T₀ 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 ₂=kT₂2dJ=1-q^-1-113q^-2-413q^-3-213445q^-4-1331415q^-5-, while for self-avoiding walks =limit of|{c₍|}^1{n}asn=1-^-2-2^-3-11^-4-62^-5-. Numerical extrapolation yields accurate estimates for ₂ and when d=2 to 6 and indicates that diverges as (T-{T₂) }^-1+ (d) where 3 (d) 4, 12, 321, 802, 18812, (d=2, 3), and that c₍n^^n (n) with 1 (d) 3, 6, 140. 3, 321. 5, 727,.