Convergence of lattice trees to super-Brownian motion above the critical dimension.

Abstract

We use the lace expansion to prove asymptotic formulae for the Fourier transforms of the r-point functions for a spread-out model of critically weighted lattice trees on the d-dimensional integer lattice for d>8. A lattice tree containing the origin defines a sequence of measures on the lattice, and the statistical mechanics literature gives rise to a natural probability measure on the collection of such lattice trees. Under this probability measure, our results, together with the appropriate limiting behaviour for the survival probability, imply convergence to super-Brownian excursion in the sense of finite-dimensional distributions.