Average Cluster Numbers in Bond Percolation on Infinitely-Length Lattice Strips

Shu-Chiuan Chang^1*, Robert Shrock²

¹Department of Physics, National Cheng Kung University, Tainan, Taiwan
²C. N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY, USA

* Presenter:Shu-Chiuan Chang, email:scchang@mail.ncku.edu.tw

We calculate exact analytic expressions for the average cluster numbers <κ>_Λs on infinite-length strips Λs of the square, triangular, and honeycomb lattices as functions of the bond occupation probability, p. Our calculations are performed for infinite-length strips with various widths and several types of transverse boundary conditions. We prove that these expressions are rational functions of p. As special cases of our results, we obtain exact values of <κ>_Λs evaluated at the critical percolation probabilities p_c,Λ for the corresponding infinite two-dimensional lattices Λ, and compare these results with an analytic finite-size correction formula. We also analyze how unphysical poles in <κ>_Λs determine the radii of convergence of series expansions for small p and for p near to unity.

Keywords: bond percolation, average cluster numbers, infinitely-length lattice strips