Percolation in the Hyperbolic Plane

Percolation in the Hyperbolic Plane

Following is a study of percolation in the hyperbolic plane H and on regular tilings in the hyperbolic plane. The processes discussed include Bernoulli site and bond percolation on planar hyperbolic graphs, invariant dependent percolations on such graphs, and Poisson-Voronoi-Bernoulli percolation. We prove the existence of three distinct nonempty phases for the Bernoulli processes. In the first...

Hard-sphere percolation: Some positive answers in the hyperbolic plane and on the integer lattice

Bootstrap percolation and kinetically constrained models on hyperbolic lattices

We study bootstrap percolation (BP) on hyperbolic lattices obtained by regular tilings of the hyperbolic plane. Our work is motivated by the connection between the BP transition and the dynamical transition of kinetically constrained models, which are in turn relevant for the study of glass and jamming transitions. We show that for generic tilings there exists a BP transition at a nontrivial cr...

An Extension of Poincare Model of Hyperbolic Geometry with Gyrovector Space Approach

‎The aim of this paper is to show the importance of analytic hyperbolic geometry introduced in [9]‎. ‎In [1]‎, ‎Ungar and Chen showed that the algebra of the group $SL(2,mathbb C)$ naturally leads to the notion of gyrogroups ‎and gyrovector spaces for dealing with the Lorentz group and its ‎underlying hyperbolic geometry‎. ‎They defined the Chen addition and then Chen model of hyperbolic geomet...

Constructing Low Degree Hyperbolic Surfaces In

We describe a new method of constructing Kobayashi-hyperbolic surfaces in complex projective 3-space based on deforming surfaces with a " hyperbolic non-percolation " property. We use this method to show that general small deformations of certain singular abelian surfaces of degree 8 are hyperbolic. We also show that a union of 15 planes in general position in projective 3-space admits hyperbol...