Percolation processes in two dimensions. V. The exponent δpand scaling theory

Abstract

For pt.IV see ibid., vol.9, p.725 (1976). By introducing a notional field variable lambda into the percolation problem, a function P_c( 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 P_c( 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.