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

Authors :
Georgakopoulos, Agelos
Panagiotis, Christoforos
Publication Year :
2021

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.

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2107.06272
Document Type :
Working Paper