Lattice trees and super-Brownian motion

http://dx.doi.org/10.4153/CMB-1997-003-8
Canad. Math. Bull. 40(1997), 19-38

  Published:1997-03-01
 Printed: Mar 1997

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 {\it 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.