We calculate exact analytic expressions for the average cluster numbers ${\langle k\rangle }_{{\mathrm{\Lambda }}_s}$ on infinite-length strips ${\mathrm{\Lambda }}_s$, 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 ${\langle k\rangle }_{{\mathrm{\Lambda }}_s}$ and derivatives of ${\langle k\rangle }_{{\mathrm{\Lambda }}_s}$ with respect to $p$, evaluated at the critical percolation probabilities $p_{c,\mathrm{\Lambda }}$ for the corresponding infinite two-dimensional lattices $\mathrm{\Lambda }$. We compare these exact results with an analytic finite-size correction formula and find excellent agreement. We also analyze how unphysical poles in ${\langle k\rangle }_{{\mathrm{\Lambda }}_s}$ 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.

• Received 21 May 2021
• Accepted 15 September 2021

DOI:https://doi.org/10.1103/PhysRevE.104.044107