Abstract
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.
Highlights
Discrete models of statistical mechanics are usually based on Euclidean lattices such as d, some researchers have studied the properties of standard models on various non-Euclidean lattices
The models from statistical mechanics that we study here are lattice animals, lattice trees, and percolation
We shall prove that these three models all exhibit “mean field” scaling behaviour on hyperbolic lattices in two and three dimensions
Summary
Discrete models of statistical mechanics are usually based on Euclidean lattices such as d (the d-dimensional hypercubic lattice), some researchers have studied the properties of standard models on various non-Euclidean lattices. In this paper we shall study properties of three standard statistical mechanical models on hyperbolic lattices. Hyperbolic lattices differ from Euclidean lattices in that the number of sites within distance N of the origin grows exponentially in N rather than polynomially. We shall prove that these three models all exhibit “mean field” scaling behaviour on hyperbolic lattices in two and three dimensions. Subsection 1.2 introduces lattice animals and lattice trees and presents our main results about them. If z0 is a specified limit point of S (possibly ∞), we write f ∼ g to mean that f (z)/g(z) converges to 1 as z → z0
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