Crossing on hyperbolic lattices
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.
Keywords
Cite
@article{arxiv.1111.5626,
title = {Crossing on hyperbolic lattices},
author = {Hang Gu and Robert M. Ziff},
journal= {arXiv preprint arXiv:1111.5626},
year = {2012}
}
Comments
Final published version, with some additions at the end