Crossing on hyperbolic lattices

TL;DR

This paper investigates percolation crossing probabilities on hyperbolic lattices, analyzing how these probabilities change with bond occupation probability and identifying critical thresholds and self-duality points.

Contribution

It provides bounds and estimates for critical percolation thresholds on various hyperbolic lattices, including {7,3} and {5,5}, and compares results with existing studies.

Findings

Crossing probability increases from 0 to 1 as p increases.

Bounds and estimates for critical thresholds p_l and p_u are established.

Self-duality point p* where crossing probability equals 1/2 is identified.

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.