Series Expansion of the Percolation Threshold on Hypercubic Lattices

Abstract

We study proper lattice animals for bond- and site-percolation on the hypercubic lattice ${\mathbb{Z}}^d$ 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 $\pcsite$ and one more term to the series for $\pcbond$, 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 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.

arXiv comment36 pages, 6 figures, 9 tables

References in corpus (7)

• Enumerations of lattice animals and trees
• Simulations of lattice animals and trees
• Percolation Thresholds and Fisher Exponents in Hypercubic Lattices
• Counting Lattice Animals in High Dimensions
• Effect of Dimensionality on the Percolation Thresholds of Various $d$-Dimensional Lattices
• Mean-field behavior for nearest-neighbor percolation in $d>10$
• Dimensional Reduction and Crossover to Mean-Field Behavior for Branched Polymers

Cited by in corpus (10)

• An Analysis of the Gel Point of Polymer Model Networks by Computer Simulations
• Bond percolation on simple cubic lattices with extended neighborhoods
• Precise bond percolation thresholds on several four-dimensional lattices
• Bond percolation between $k$ separated points on a square lattice
• Exact Percolation Probability on the Square Lattice
• Dynamics around the Site Percolation Threshold on High-Dimensional Hypercubic Lattices
• Percolation thresholds on high dimensional $D_n$ and dense packing lattices
• Critical site percolation in high dimension
• Extended-range percolation in five dimensions
• Expansion of the Critical Intensity for the Random Connection Model