2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/204871Invariant 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.15 pages. Some changes in introduction and discussion; to appear in J. Math. PhysMathematical PhysicsHigh Energy Physics - LatticeHigh Energy Physics - Theory41A60; 82B80text