NoBLE for lattice trees and lattice animals
Robert Fitzner, Remco van der Hofstad

TL;DR
This paper applies the non-backtracking lace expansion (NoBLE) method to lattice trees and animals on $\,\mathbb{Z}^d$, establishing mean-field behavior above certain high dimensions and refining the critical dimension bounds.
Contribution
It develops a non-backtracking lace expansion for lattice trees and animals, providing sharper bounds and extending the dimension range where mean-field behavior is proven.
Findings
Mean-field behavior for lattice trees above dimension 16
Mean-field behavior for lattice animals above dimension 17
Refined bounds on the critical dimension for these models
Abstract
We study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension and , 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 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 for…
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