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Lattice trees and super-Brownian motion

  Published:1997-03-01
 Printed: Mar 1997
  • Eric Derbez
  • Gordon Slade
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Abstract

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 {\it 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.
MSC Classifications: 82B41, 60K35, 60J65 show english descriptions Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Brownian motion [See also 58J65]
82B41 - Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]
60K35 - Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
60J65 - Brownian motion [See also 58J65]
 

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