Series Expansion of Critical Densities for Percolation on $\mathbb{Z}^d$

Stephan Mertens, Cristopher Moore

Published 2018-05-07Version 1

We study proper lattice animals for bond- and site-percolation on the hypercubic lattice $\mathbb{Z}^d$ to derive asymptotic series of the critical densities $p_c$ in $1/d$, The first few terms of these series were computed in the 1970s, but the series have not been extended since then. We add two more terms to the series for $\pcsite$ and one more term to the series for $\pcbond$, using a combination of brute-force enumeration, combinatorial identities and an approach based on Pad\'e approximants, which requires much less resources than the classical method. We discuss why it took 40 years to compute these terms, and what it would take to compute the next ones. En passant, we present new perimeter polynomials for site and bond percolation and numerical values for the growth rate of bond animals.