Eric Derbez, Gordon Slade.
Lattice trees and super-Brownian motion.
(63K, tex)
ABSTRACT. This article discusses our recent proof that above eight dimensions
the scaling limit of sufficiently spread-out
lattice trees is the variant of super-Brownian motion called
integrated super-Brownian excursion (ISE),
as conjectured by Aldous.
The same is true for nearest-neighbour lattice trees in sufficiently
high dimensions.
The proof, whose details will appear elsewhere,
uses the lace expansion. Here, a related but simpler analysis is applied to
show that the scaling limit of a mean-field theory is ISE, in all dimensions.
A connection is drawn between ISE and
certain generating functions and
critical exponents, which may be useful for the
study of high-dimensional percolation models at the critical point.