Exact results for average cluster numbers in bond percolation on infinite-length lattice strips

We calculate exact analytic expressions for the average cluster numbers (k)As on infinite-length strips As, 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 (k)As and derivatives of (k)As with respect to p, evaluated at the critical percolation probabilities pc,A for the corresponding infinite two-dimensional lattices A. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in (k)As 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.

Automated summary

In this study, exact analytic expressions for the average cluster numbers on infinite-length strips of different lattices as functions of bond occupation probability were calculated. These expressions were proven to be rational functions of p. The comparison with an analytic finite-size correction formula showed excellent agreement, and the analysis of unphysical poles determined the radii of convergence of series expansions.