Critical Exponents in Percolation via Lattice Animals
Alan Hammond, U.C. Berkeley, USA
Abstract
We examine the percolation model on $mathbb{Z}^d$ by an approach
involving lattice animals and their surface-area-to-volume ratio.
For $beta in [0,2(d-1))$, let
$f(beta)$ be the asymptotic exponential rate in the number
of edges of the number of lattice animals containing the origin which
have surface-area-to-volume ratio $beta$. The function $f$ is bounded
above by a function which may be written in an explicit form. For low
values of $beta$ (mbox{$beta leq 1/p_c - 1$}), equality holds, as
originally demonstrated by F.Delyon. For higher values ($beta > 1/p_c
- 1$), the inequality is strict.
We introduce two critical exponents, one of which describes how
quickly $f$ falls away from the explicit form as $beta$ rises from
$1/p_c - 1$, and the second of which describes how large clusters
appear in the marginally subcritical regime of the percolation model.
We demonstrate that the pair of exponents must satisfy certain
inequalities. Other such inequalities yield sufficient
conditions for the absence of an infinite cluster at the critical
value (c.f. cite{techrep}).
The first exponent is related to one of a more conventional nature in
the scaling theory of percolation, that of correlation size. In
deriving this relation, we find that there are two possible
behaviours, depending on the value of the first exponent, for the
typical surface-area-to-volume ratio of an unusually large cluster in
the marginally subcritical regime.
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