Continuous vs discrete spins in the hyperbolic plane

TL;DR

This paper investigates the $O(n)$ model on graphs resembling the hyperbolic plane, revealing that pair correlations decay exponentially at all temperatures only when $n>1$, highlighting a phase transition related to the spin type.

Contribution

The study provides new insights into the behavior of the $O(n)$ model on hyperbolic-like graphs, especially regarding correlation decay and phase transitions for different $n$ values.

Findings

Exponential decay of correlations occurs for all temperatures if and only if n>1.

The model exhibits distinct behaviors depending on the value of n.

The geometry of the underlying graph influences the correlation properties.

Abstract

We study the $O(n)$ model on graphs quasi-isometric to the hyperbolic plane, with free boundary conditions. We observe that the pair correlations decay exponentially with distance, for all temperatures, if and only if $n>1$.