Modèle de dimère sur le tore et bijection de Temperley

Temperley's bijection relates the toroidal dimer model to cycle rooted spanning forests (CRSF) on the torus. The height function of the dimer model and the homology class of CRSF are naturally related. When the size of the torus tends to infinity, we show that the measure on CRSF arising from the dimer model converges to a measure on (disconnected) spanning forests or spanning trees. There is a phase transition, which is determined by the average height change.