Studies in percolation theory

• Main Author: Blease, John Hedley James
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-1976
• Subjects: Condensed matter physics; Theoretical physics
• ISBN: 0355687275; 9780355687279

We consider percolation problems on lattices in which bonds are or are not directed, conveniently referred to as directed and undirected lattices. For the undirected lattices our main results are estimates of the exponent [beta] describing the vanishing of the percolation probability. We obtain the estimates. We derive what are believed to be the first series expansion results for certain directed lattices. As well as estimating [beta], we have also estimated the exponents [gamma]n associated with the spherical moments [mu]n(p) of the pair-connectedness. Within the limits of uncertainty it is found that [gamma]n is of the form [gamma]n = [gamma] + nv, where [gamma] is the exponent describing the divergence of the mean-size. Our estimates areLower bounds for p , the critical probability, are also established.In the directed case, we also investigate the shape of the clusters bycalculating the pair-connectedness of a lattice site whose polar coordinates are r and [theta], for certain values of r and [theta]. By postulating an asymptotic form for the pair-connectedness we derive a relation between certain exponents. Two methods for deriving series expansions for S(p) and pc for directed simple hypercubic lattices (of coordination number z = 2d, d being the dimensionality) are also described. The expansion we derive for pc is given byThe possible existence of a critical dimensionality, above which the exponents assume their "mean-field" values, is investigated numerically. Although our estimates of [gamma] appear to approach the limit [gamma] = 1 smoothly, due to numerical uncertainty in [gamma] for the larger values of d it is not possible to draw any firm conclusions.Computing techniques for various mapping problems which arose during the course of our work are also discussed.