On the exponential growth rates of lattice animals and interfaces II: new asymptotic bounds

TL;DR

This paper develops a method linking percolation thresholds and lattice animal growth rates, improving asymptotic bounds for high-dimensional lattices and providing new bounds for 3D percolation.

Contribution

It introduces a novel approach connecting percolation thresholds with lattice animal growth, leading to improved bounds and new results in high-dimensional lattice models.

Findings

Improved asymptotic bounds on lattice animal growth rates as dimension increases.

Established a lower bound for the percolation threshold in 3D.

Created a method to translate bounds between percolation thresholds and growth rates.

Abstract

We introduce a method for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate $a(G)$ of lattice animals and vice-versa. We exploit this in both directions. We improve on the best known asymptotic lower and upper bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. We use percolation as a tool to obtain the latter, and conversely we use the former to obtain lower bounds on $p_c(\mathbb{Z}^d)$. We obtain the rigorous lower bound $\dot{p}_c(\mathbb{Z}^3)>0.2522$ for 3-dimensional site percolation.