Constraint percolation on hyperbolic lattices

TL;DR

This paper investigates four types of constraint percolation models on hyperbolic lattices, revealing that $k$-core models behave similarly to ordinary percolation while force-balance percolation exhibits a discontinuous transition, supported by numerical and rigorous analysis.

Contribution

It provides the first comparative study of multiple constraint percolation models on hyperbolic lattices, including rigorous proofs and improved numerical methods.

Findings

$k$-core models show behavior similar to ordinary percolation.

Force-balance percolation transition is discontinuous.

Existence of a critical probability less than one for force-balance percolation.

Abstract

Hyperbolic lattices interpolate between finite-dimensional lattices and Bethe lattices and are interesting in their own right with ordinary percolation exhibiting not one, but two, phase transitions. We study four constraint percolation models---$k$-core percolation (for $k=1,2,3$) and force-balance percolation---on several tessellations of the hyperbolic plane. By comparing these four different models, our numerical data suggests that all of the $k$-core models, even for $k=3$, exhibit behavior similar to ordinary percolation, while the force-balance percolation transition is discontinuous. We also provide a proof, for some hyperbolic lattices, of the existence of a critical probability that is less than unity for the force-balance model, so that we can place our interpretation of the numerical data for this model on a more rigorous footing. Finally, we discuss improved numerical methods for determining the two critical probabilities on the hyperbolic lattice for the $k$-core percolation models.