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Trees, animals, and percolation on hyperbolic lattices. (English) Zbl 1226.60133

Summary: We study lattice trees, lattice animals, and percolation on non-Euclidean lattices that correspond to regular tessellations of two- and three-dimensional hyperbolic space. We prove that critical exponents of these models take on their mean field values. Our methods are mainly combinatorial and geometric.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
05B45 Combinatorial aspects of tessellation and tiling problems
51M09 Elementary problems in hyperbolic and elliptic geometries
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
82B43 Percolation
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