NoBLE for lattice trees and lattice animals

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 $\mathbb{Z}^d$ in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension $16$ and $17$, 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 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 $d_c=8$ 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) to derive a non-backtracking lace expansion for the LT and LA two-point functions; (b) to bound the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) to obtain 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.