D S Gaunt

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.

Bond percolation processes in d dimensions

The authors study bond percolation processes on a d-dimensional simple hypercubic lattice. 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 fourth order, respectively. The zeroth-order terms are the Bethe approximations. The critical probability pc is found to have the expansion, probably asymptotic, pc= sigma -1(1+21/2 sigma -2+71/2 sigma -3+57 sigma -4+...), while the cluster growth parameter lambda can be expanded as lambda = lambda B(1-2 sigma -2-...) where lambda B is the Bethe approximation for lambda . They also present series data for the mean cluster size and the cluster growth function for d=4 to 7. Numerical analysis suggests that the critical dimension, dc, for bond percolation is dc=6, as it seems to be for the site problem. The evidence also supports the conjecture that the value of a particular critical exponent in a given dimension is the same for both bond and site processes.

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.

Universality in branched polymers on d-dimensional hypercubic lattices

The authors have derived series for weakly and strongly embeddable trees in d-dimensional simple hypercubic lattices for arbitrary integral d. For d=2,3,...,9 they present series evidence that such trees are in the same universality class as lattice animals. In addition they have derived expansions in inverse powers of sigma =2d-1 for the growth parameters for bond and site trees and compare these with the corresponding results for animals.

The critical dimension for lattice animals

Recent field theoretical calculations for lattice animals by Lubensky and Isaacson (see Phys. Rev. A, vol.20, p.2130, 1979) yield dc=8 as the critical dimension and provide a first order epsilon -expansion for the exponent theta . Support for these predictions may be obtained by extending the previous work of Gaunt and Rushkin (see ibid., vol.11, p.1369, 1978) on the exact enumeration of site and bond animals on a d-dimensional simple hypercubic lattice to arbitrary d.

Lattice models of branched polymers: statistics of uniform stars

The authors investigate the statistical properties of uniform star polymers with f branches, modelled on lattices in two and three dimensions. It is shown that the growth constant exists and is equal to mu f, where mu is the self-avoiding walk limit. The f dependence of the corresponding critical exponent gamma (f) is studied using exact enumeration and Monte Carlo techniques and the results are compared with the predictions of Miyake and Freed (1983) obtained using chain conformation space renormalisation group method.

Percolation processes in two dimensions. V. The exponent δp and scaling theory

For pt.IV see ibid., vol.9, p.725 (1976). By introducing a notional field variable lambda into the percolation problem, a function Pc( lambda ) is defined whose Ising analogue is the magnetic field variation of the magnetization along the critical isotherm. Series expansions are used to study the critical behaviour of Pc( lambda ) characterized by an exponent delta p, for both site and bond percolation problems on the more common two-dimensional lattices. The authors conclude that delta p is a dimensional invariant and estimate delta p=18.0+or-0.75. It appears that delta p=18, lambda p=23/7, beta p=1/7 is the simplest set of rational exponents which is most consistent with the available data and which satisfies the scaling law gamma p= beta p( delta p-1) exactly.

On the asymptotic number of lattice animals in bond and site percolation

A recently proposed asymptotic form, due to Domb (1976), for the total number of bond and site animals of size n, has been investigated numerically. It is found to fit the available data better than simpler forms previously assumed. The critical parameters entering into the asymptotic form are estimated for a number of two- and three-dimensional lattices, and conclusions are drawn about their lattice and dimensional dependence. In particular, the cluster growth parameter lambda is estimated with a higher degree of precision than that previously attained.