On the large N expansion in hyperbolic sigma-models

Abstract

Invariant correlation functions for $\mathrm{SO}(1,N)$ hyperbolic sigma-models are investigated. The existence of a large $N$ asymptotic expansion is proven on finite lattices of dimension $d\ge 2$. 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.

Comments: 15 pages. Some changes in introduction and discussion; to appear in J. Math. Phys