Exact Results for Average Cluster Numbers in Bond Percolation on Infinite-Length Lattice Strips

TL;DR

This paper derives exact rational formulas for the average number of clusters in bond percolation on infinite-length lattice strips, providing insights into critical probabilities and series convergence.

Contribution

It presents the first exact analytic expressions for average cluster numbers on infinite-length lattice strips across various lattices and boundary conditions, linking them to critical percolation thresholds.

Findings

Exact rational functions for average cluster numbers as functions of $p$.

Validation of formulas through comparison with finite-size correction models.

Analysis of unphysical poles affecting series convergence.

Abstract

We calculate exact analytic expressions for the average cluster numbers $\langle k \rangle_{\Lambda_s}$ on infinite-length strips $\Lambda_s$, with various widths, of several different lattices, as functions of the bond occupation probability, $p$. It is proved that these expressions are rational functions of $p$. As special cases of our results, we obtain exact values of $\langle k \rangle_{\Lambda_s}$ and derivatives of $\langle k \rangle_{\Lambda_s}$ with respect to $p$, evaluated at the critical percolation probabilities $p_{c,\Lambda}$ for the corresponding infinite two-dimensional lattices $\Lambda$. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in $\langle k \rangle_{\Lambda_s}$ determine the radii of convergence of series expansions for small $p$ and for $p$ near to unity. Our calculations are performed for infinite-length strips of the square, triangular, and honeycomb lattices with several types of transverse boundary conditions.