On the large N expansion in hyperbolic sigma-models

TL;DR

This paper proves the existence of a large N asymptotic expansion for invariant correlation functions in hyperbolic sigma-models on finite lattices, overcoming technical challenges with novel coordinate and combinatorial methods.

Contribution

It establishes the large N expansion for hyperbolic sigma-models and characterizes the saddle point, using horospherical coordinates and the matrix-tree theorem.

Findings

Existence of large N asymptotic expansion proven

Unique saddle point with negative gap identified

Technical methods adapted for non-compact case

Abstract

Invariant correlation functions for ${\rm SO}(1,N)$ hyperbolic sigma-models are investigated. The existence of a large $N$ asymptotic expansion is proven on finite lattices of dimension $d \geq 2$. The unique saddle point configuration is characterized by a negative gap vanishing at least like 1/V with the volume. Technical difficulties compared to the compact case are bypassed using horospherical coordinates and the matrix-tree theorem.