Constraint percolation on hyperbolic lattices

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.