M F Sykes

Exact Critical Percolation Probabilities for Site and Bond Problems in Two Dimensions

An exact method for determining the critical percolation probability, pc, for a number of two‐dimensional site and bond problems is described. For the site problem on the plane triangular lattice pc = ½. For the bond problem on the triangular, simple quadratic, and honeycomb lattices, pc=2 sin (118π),12,1−2 sin (118π), respectively. A matching theorem for the mean number of finite clusters on certain two‐dimensional lattices, somewhat analogous to the duality transformation for the partition function of the Ising model, is described.

Use of Series Expansions for the Ising Model Susceptibility and Excluded Volume Problem

The present paper discusses the problem of making the most effective use of the coefficients of series expansions for the Ising model and excluded volume problem in estimating critical behavior. It is shown that after initial irregularities the coefficients appear to settle down to a smooth asymptotic behavior. Alternative methods of analysis are considered for the provision of a steady series of approximations to the critical point. Numerical conclusions are drawn for particular lattices for which additional terms have recently become available.

Derivation of Low‐Temperature Expansions for the Ising Model of a Ferromagnet and an Antiferromagnet

Low‐temperature expansions for the free energy of the Ising model of a ferromagnet and an antiferromagnet are derived for the more usual two‐ and three‐dimensional lattices. The underlying enumerative problem is studied and a new method described that makes it possible to obtain more terms than available previously without undue labor.

Percolation processes in two dimensions. I. Low-density series expansions

The derivation of low-density series expansions for the mean cluster size in random site and bond mixtures on a two-dimensional lattice is described briefly. New data are given for the triangular, simple quadratic and honeycomb lattices.

Some Counting Theorems in the Theory of the Ising Model and the Excluded Volume Problem

The problem of the exact enumeration of self‐avoiding random walks on a lattice is studied and a theorem derived that enables the number of such walks to be calculated recursively from the number of a restricted class of closed graphs more easily enumerated than the walks themselves. The method of Oguchi for deriving a high‐temperature expansion for the zero‐field susceptibility of the Ising model is developed and a corresponding theorem enabling the successive coefficients to be calculated recursively from a restricted class of closed graphs deduced. The theorem relates the susceptibility to the configurational energy and enables the behavior of the antiferromagnetic susceptibility at the transition point to be inferred.

Percolation processes in three dimensions

The derivation of low-density series expansions for the mean cluster size in random site and bond mixtures on a three-dimensional lattice is described briefly. New data are given for the face-centred cubic, body-centred cubic, simple cubic and diamond lattices. The critical concentrations for the site problem is estimated as pc=0.198+or-0.003 (FCC), pc=0.245+or-0.004 (BCC), pc=0.310+or-0.004 (SC), pc=0.428+or-0.004 (D); for the bond problem as pc=0.119+or-0.001 (FCC), pc=0.1785+or-0.002 (BCC), pc=0.247+or-0.003 (SC), pc=0.388+or-0.005 (D). It is concluded that the data are reasonably consistent with the hypothesis that the mean cluster size S(p) approximately=C(pc-p)- gamma as p to pc-with gamma a dimensional invariant, gamma =1.66+or-0.07 in three dimensions. Estimates of the critical amplitude C are also given.

Percolation processes in d-dimensions

Series data for the mean cluster size for site mixtures on a d-dimensional simple hypercubical lattice are presented. Numerical evidence for the existence of a critical dimension for the cluster growth function and for the mean cluster size is examined and it is concluded that dc=6. Exact expansions for the mean number of clusters K(p) and the mean cluster size S(p) in powers of 1/ sigma where sigma =2d-1 and p<pc are derived through fifth and third order, respectively. The zeroth-order terms are the Bethe approximations.

Susceptibility and fourth-field derivative of the spin-1/2 Ising model for T>Tc and d=4

THe authors investigate the spin-1/2 Ising model with nearest-neighbour interactions on the four-dimensional simple hypercubic lattice. High-temperature series expansions are studied for the zero-field susceptibility chi 0 and the fourth-field derivative of the free energy Xi 0(2) up to order nu 17. The series are analysed for singularities of the form t-1 mod 1nt mod p where t is the reduced temperature. For chi 0 it is found that p=0.33+or-0.07 when q=1, in good agreement with the prediction p=1/3, q=1 of renormalisation group theory. The critical temperature is estimated to be nu c-1=6.7315+or-0.0015. Results for chi 0(2) are more slowly convergent but are not inconsistent with the renormalisation group prediction p=1/3, q=4.

Percolation processes in two dimensions. II. Critical concentrations and the mean size index

For pt.I see ibid., vol.9, p.87 (1976). New series data are examined for the mean cluster size for site and bond mixtures in two dimensions. The critical concentration for the site problem on the simple quadratic lattice is estimated as pc=0.593+or-0.002 and on the honeycomb lattice as pc=0.698+or-0.003. It is concluded that the data are reasonably consistent with the hypothesis that the mean cluster size S(p) approximately=C(pc-p)- gamma as p to pc- with gamma a dimensional invariant, gamma =2.43+or-0.03 in two dimensions. Estimates of the critical amplitude C are also given.

Derivation of low‐temperature expansions for Ising model. III. Two‐dimensional lattices‐field grouping

The derivation of series expansions appropriate for low temperatures or high applied magnetic fields for the two‐dimensional Ising model of a ferromagnet and antiferromagnet is studied as a field grouping. New results are given for the high field polynomials for the triangular lattice to order 10, the simple quadratic lattice to order 15, and the honeycomb lattice to order 21.