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

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 $p=1$.