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Lattice trees and super-Brownian motion

Published online by Cambridge University Press:  20 November 2018

Eric Derbez  and
Gordon Slade
Eric Derbez
Affiliation:
Department of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1 e-mail: slade@mcmaster.caderbez@math.ubc.ca
Gordon Slade
Affiliation:
Department of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1 e-mail: slade@mcmaster.caderbez@math.ubc.ca
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Abstract

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This article discusses our recent proof that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion called integrated super-Brownian excursion (ISE), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof, whose details will appear elsewhere, uses the lace expansion. Here, a related but simpler analysis is applied to show that the scaling limit of a mean-field theory is ISE, in all dimensions. A connection is drawn between ISE and certain generating functions and critical exponents, which may be useful for the study of high-dimensional percolation models at the critical point.

Keywords

Primary: 82B4160K3560J65
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 1 , 01 March 1997 , pp. 19 - 38
Copyright
Copyright © Canadian Mathematical Society 1997

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