2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/186221We compute in small temperature expansion the two-loop renormalization constants and the three-loop coefficient of the beta-function, that is the first non-universal term, for the sigma-model with O(N) invariance on the triangular lattice at N=-1. The partition function of the corresponding Grassmann theory is, for negative temperature, the generating function of unrooted forests on such a lattice, where the temperature acts as a chemical potential for the number of trees in the forest. To evaluate Feynman diagrams we extend the coordinate space method to the triangular lattice.26 pages, 4 figuresStatistical MechanicsHigh Energy Physics - Latticetext