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.
Mark Holmes. "Convergence of Lattice Trees to Super-Brownian Motion above the Critical Dimension." Electron. J. Probab. 13 671 - 755, 2008. https://doi.org/10.1214/EJP.v13-499