The Lace Expansion for Lattice Trees

AbstractWe study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice $${\mathbb {Z}}^d$$ Z d in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension $$16$$ 16 and $$17$$ 17 , respectively. Such results have previously been obtained by Hara and Slade in sufficiently high dimensions. The dimension above which their results apply was not yet specified. We rely on the non-backtracking lace expansion (NoBLE) method that we have recently developed. The NoBLE makes use of an alternative lace expansion for LAs and LTs that perturbs around non-backtracking random walk rather than around simple random walk, leading to smaller corrections. The NoBLE method then provides a careful computational analysis that improves the dimension above which the result applies. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to $$d_c=8$$ d c = 8 for both models, as is known for sufficiently spread-out models by the results of Hara and Slade mentioned earlier. The main ingredients in this paper are (a) a derivation of a non-backtracking lace expansion for the LT and LA two-point functions; (b) bounds on the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) sharp numerical bounds on the coefficients. Our proof is complemented by a computer-assisted numerical analysis that verifies that the necessary bounds used in the NoBLE are satisfied.

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Lattice trees and super-Brownian motion

Canadian Mathematical Bulletin ◽

10.4153/cmb-1997-003-8 ◽

1997 ◽

Vol 40 (1) ◽

pp. 19-38 ◽

Cited By ~ 7

Author(s):

Eric Derbez ◽

Gordon Slade

Keyword(s):

Brownian Motion ◽

Mean Field Theory ◽

Scaling Limit ◽

Mean Field ◽

High Dimensional ◽

High Dimensions ◽

Lace Expansion ◽

Brownian Excursion ◽

Lattice Trees ◽

Super Brownian Motion

AbstractThis 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 calledintegrated 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.

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