NoBLE for Lattice Trees and Lattice Animals

Robert Fitzner,Remco Van Der Hofstad

doi:10.1007/s10955-021-02816-z

Robert Fitzner, Remco Van Der Hofstad

Open Access

https://doi.org/10.1007/s10955-021-02816-z

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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 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 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.

Highlights

1.1 MotivationLattice trees (LTs) and lattice animals (LAs) are models for branched polymers, where excluded volume creates a self-avoidance interaction between different pieces, while the branching nature corresponds to polymers whose building blocks can have covalent bonds to more than two other building blocks
Like many statistical physics models, lattice trees (LTs) and LAs are expected to have a so-called upper critical dimension, above which their behavior should be similar to a simpler model having less intricate interactions
We prove that the LT and LA two-point functions satisfy a so-called infra-red bound that describes its singularity for small Fourier variables

Summary

Introduction

Lattice trees (LTs) and lattice animals (LAs) are models for branched polymers, where excluded volume creates a self-avoidance interaction between different pieces (vertices or bonds), while the branching nature corresponds to polymers whose building blocks can have covalent bonds to more than two other building blocks. They are the branching equivalents of self-avoiding walks (see [33] for a detailed introduction, and a brief introduction to LTs and LAs as well). For LTs and LAs, the upper critical dimension is believed to be dc = 8, while this simpler model is branching random walk (BRW). See for example Perkins [39] for super-process limits of BRWs

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