Continuous Versus Discrete Spins in the Hyperbolic Plane

Abstract

Abstract. We study the O(n) model on graphs quasi-isometric to the hyperbolic plane, with free boundary conditions. We observe that the pair correlation decays exponentially with distance, for all temperatures, if and only if n > 1.We wish to report here on a curious contrast between the behaviour of models with discrete and continuous symmetry on graphs quasi-isometric to the hyperbolic plane. For concreteness we will study the O(n) model, with n = 1, or Ising, being our model for a model with discrete spins and n ≥ 2 our model for continuous spins and symmetry group. Let us start with the relevant definitions.Definition. Two metric spaces X and Y are called quasi-isometric if there is a map φ : X → Y with the propertieswhere c and C denote constants which are independent of x and y.We say that a graph is quasi-isometric to the hyperbolic plane if, when you equip it with the graph metric (i.e. the distance between any two vertices is the length of the shortest path between them), it is quasi-isometric to the hyperbolic plane with its standard metric.An easy example of a graph quasi-isometric to the hyperbolic plane is the seven-regular planar triangulation: the (unique) graph which is the 1-skeleton of an infinite triangulation of a planar disk which satisfies that the degree of every vertex is 7 (the number 7 may be replaced with any n > 6 and the graph would still be quasi-isometric to the hyperbolic plane). This