Crossing on hyperbolic lattices
arXiv:1111.5626 · doi:10.1103/PhysRevE.85.051141
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 {5,5} (pentagonal) lattices. We find that the crossing probability increases gradually from zero to one 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.
Final published version, with some additions at the end
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- Renormalization group for link percolation on planar hyperbolic manifolds
- Site-Percolation Threshold of Carbon Nanotube Fibers: Fast Inspection of Percolation with Markov Stochastic Theory
- Percolation Thresholds in Hyperbolic Lattices
- Bounds of percolation thresholds on hyperbolic lattices
- A homological upper bound on critical probabilities for hyperbolic percolation
- Constraint percolation on hyperbolic lattices
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