Constraint percolation on hyperbolic lattices

Jorge H. Lopez, J. M. Schwarz
2017 Physical review. E  
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
more » ... 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.
doi:10.1103/physreve.96.052108 pmid:29347694 fatcat:i5wspr5t2vehzd5mgviiwztn6q