Series expansion of the percolation threshold on hypercubic lattices. (25th October 2018)

• Record Type: Journal Article
• Authors: Mertens, Stephan; Moore, Cristopher
• Is Part Of: Journal of physics. Volume 51:Number 47(2018)
• Publication Date: 2018-10-25
• Subjects: percolation -- asymptotic series -- cubic lattice -- low-density expansion -- high-dimensional expansion; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aae65c
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We study proper lattice animals for bond- and site-percolation on the hypercubic lattice to derive asymptotic series of the percolation threshold 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 and one more term to the series for, using a combination of brute-force enumeration, combinatorial identities and an approach based on Padé approximants, which requires much fewer resources than the classical method. We discuss why it took 40 yr 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.