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 in Zd for d > 8. A lattice tree containing the origin defines a sequence of measures on Zd, 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 lim- iting behaviour for the survival probability, imply convergence to super-Brownian excursion in the sense of finite-dimensional distributions . © 2008 Applied Probability Trust.