Crossing on hyperbolic lattices

2011/11/30 by Hang Gu, Robert M. Ziff
Mathematics · Physics and Astronomy · #Binary number #Boundary (topology) #Combinatorics #Duality (order theory) #Function (biology) #Lattice (music) #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Percolation (cognitive psychology) #Physics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.dis-nn

paper · pdf · doi:10.1103/physreve.85.051141

published as Physical Review E 85, 051141 (May 29, 2012) · Final published version, with some additions at the end

openalex publication_date 2012/05/29 · arxiv created 2012/06/05 · arxiv updated 2012/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We divide the circular boundary of a hyperbolic lattice into four equal intervals and study the probability of a percolation crossing between an opposite pair as a function of the bond occupation probability p. We consider the 7,3 (heptagonal), enhanced or extended binary tree (EBT), the EBT-dual, and the 5,5 (pentagonal) lattices. We find that the crossing probability increases gradually from 0 to 1 as p increases from the lower p_l to the upper p_u critical values. We find bounds and estimates for the values of p_l and p_u for these lattices and identify the self-duality point p corresponding to where the crossing probability equals 1/2. Comparison is made with recent numerical and theoretical results.

Citations

• A fast Monte Carlo algorithm for site or bond percolation
• Percolation on hyperbolic lattices
• Critical percolation in finite geometries
• Efficient Monte Carlo Algorithm and High-Precision Results for Percolation