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 kagomé 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 $p=1$.

arXiv comment16 pages, revtex, 7 eps figures

References in corpus (22)

• Asymptotic Limits and Zeros of Chromatic Polynomials and Ground State Entropy of Potts Antiferromagnets
• Exact Potts Model Partition Functions on Ladder Graphs
• Universality of finite-size corrections to the number of critical percolation clusters
• Complex-Temperature Singularities of the Susceptibility in the $d=2$ Ising Model. I. Square Lattice
• Exact Potts Model Partition Functions on Wider Arbitrary-Length Strips of the Square Lattice
• Exact Potts Model Partition Function on Strips of the Triangular Lattice
• Determination of the bond percolation threshold for the Kagome lattice
• Some New Results on Complex-Temperature Singularities in Potts Models on the Square Lattice
• Study of the Potts Model on the Honeycomb and Triangular Lattices: Low-Temperature Series and Partition Function Zeros
• Exact Potts Model Partition Function for Strips of the Square Lattice
• Complex-Temperature Properties of the 2D Ising Model with $\beta H=\pm i\pi /2$
• Complex-Temperature Singularities in the $d=2$ Ising Model. II. Triangular Lattice
• Complex-Temperature Singularities in the $d=2$ Ising Model. III. Honeycomb Lattice
• Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
• Exact Potts Model Partition Functions for Strips of the Triangular Lattice
• Complex-Temperature Phase Diagrams for the q-State Potts Model on Self-Dual Families of Graphs and the Nature of the $q\to \infty$ Limit
• A Connection Between Complex-Temperature Properties of the 1D and 2D Spin $s$ Ising Model
• Zeros of the Partition Function for Higher--Spin 2D Ising Models
• Zeros of Jones Polynomials for Families of Knots and Links
• Complex-Temperature Partition Function Zeros of the Potts Model on the Honeycomb and Kagomé Lattices
• Potts Model Partition Functions for Self-Dual Families of Strip Graphs
• Exact Partition Function for the Potts Model with Next-Nearest Neighbor Couplings on Strips of the Square Lattice

Cited by in corpus (5)

• Predictions of bond percolation thresholds for the kagomé and Archimedean $(3,12^2)$ lattices
• Exact densities of loops in O(1) dense loop model and of clusters in critical percolation on a cylinder II: rotated lattice
• On the number of clusters for planar graphs
• Exact Results for Average Cluster Numbers in Bond Percolation on Infinite-Length Lattice Strips
• Errata and Addenda to Mathematical Constants