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2008 Convergence of Lattice Trees to Super-Brownian Motion above the Critical Dimension
Mark Holmes
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Electron. J. Probab. 13: 671-755 (2008). DOI: 10.1214/EJP.v13-499


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.


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Mark Holmes. "Convergence of Lattice Trees to Super-Brownian Motion above the Critical Dimension." Electron. J. Probab. 13 671 - 755, 2008.


Accepted: 18 April 2008; Published: 2008
First available in Project Euclid: 1 June 2016

zbMATH: 1187.82049
MathSciNet: MR2399294
Digital Object Identifier: 10.1214/EJP.v13-499

Primary: 82B41
Secondary: 60F05 , 60G57 , 60K35

Keywords: Lace expansion , Lattice trees , Super-Brownian motion

Vol.13 • 2008
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