Percolation on hyperbolic lattices

TL;DR

This paper investigates percolation transitions on hyperbolic lattices, revealing two distinct thresholds with unique properties, and suggests a new universality class different from Cayley trees.

Contribution

It provides the first numerical verification of two percolation thresholds on hyperbolic lattices and characterizes their finite-size scaling and critical exponents.

Findings

Two percolation thresholds are identified.

The lower threshold transition is similar to Cayley trees.

The upper threshold transition has distinct scaling properties.

Abstract

The percolation transitions on hyperbolic lattices are investigated numerically using finite-size scaling methods. The existence of two distinct percolation thresholds is verified. At the lower threshold, an unbounded cluster appears and reaches from the middle to the boundary. This transition is of the same type and has the same finite-size scaling properties as the corresponding transition for the Cayley tree. At the upper threshold, on the other hand, a single unbounded cluster forms which overwhelms all the others and occupies a finite fraction of the volume as well as of the boundary connections. The finite-size scaling properties for this upper threshold are different from those of the Cayley tree and two of the critical exponents are obtained. The results suggest that the percolation transition for the hyperbolic lattices forms a universality class of its own.