Percolation and lattice animals: exponent relations, and conditions for $\theta(p_c)=0$

Abstract

We examine the percolation model in $\mathbb{Z}^d$ by an approach involving lattice animals, in which their relevant characteristic is surface-area-to-volume ratio. Two critical exponents are introduced. The first is related to the growth rate in size of the number of lattice animals up to translation whose surface-area-to-volume ratio is marginally greater than $1/p_c -1$. The second describes how unusually large clusters form in the percolation model at parameter values slightly below $p_c$. Certain inequalities on the pair of exponents cannot be satisfied, while others imply the continuity of the percolation probability. The first exponent is related to one of a more conventional nature, that of correlation size. In this paper, the central aspects of the approach are described, and the proofs of the main results are presented. The report located at math.PR/0402026 gives complete proofs of all of the assertions.