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Convergence of lattice trees to super-Brownian motion above the critical dimension. (English) Zbl 1187.82049

Summary: 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.

MSC:

82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
60F05 Central limit and other weak theorems
60G57 Random measures
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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