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

TL;DR

This paper provides exact calculations of the average number of connected clusters per site in bond percolation on lattice strips, analyzing how these values approach the 2D lattice limit and exploring their complex singularities.

Contribution

It offers the first exact results for $<k>$ on finite-width lattice strips for various lattices and boundary conditions, advancing understanding of percolation on quasi-1D systems.

Findings

Exact formulas for $<k>$ on lattice strips

Analysis of $<k>$ convergence to 2D lattice values

Study of singularities in the complex $p$ plane

Abstract

We present exact calculations of the average number of connected clusters per site, $<k>$, as a function of bond occupation probability $p$, for the bond percolation problem on infinite-length strips of finite width $L_y$, of the square, triangular, honeycomb, and kagom\'e lattices $\Lambda$ with various boundary conditions. These are used to study the approach of $<k>$, for a given $p$ and $\Lambda$, to its value on the two-dimensional lattice as the strip width increases. We investigate the singularities of $<k>$ in the complex $p$ plane and their influence on the radii of convergence of the Taylor series expansions of $<k>$ about $p=0$ and $p=1$.