Dimers on Riemann surfaces II: conformal invariance and scaling limit

Nathanaël Berestycki (1) , Benoit Laslier (2, 3) , Gourab Ray (4)

1 University of Vienna [Vienna]
2 UFR Mathématiques UPCité - UFR Mathématiques [Sciences] - Université Paris Cité
3 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
4 UVIC - University of Victoria [Canada]

Given a bounded Riemann surface $M$ of finite topological type, we show the existence of a universal and conformally invariant scaling limit for the Temperleyan cycle-rooted spanning forest on any sequence of graphs which approximate $M$ in a reasonable sense (essentially, the invariance principle holds and the walks satisfy a crossing assumption). In combination with the companion paper arxiv:1908.00832, this proves the existence of a universal, conformally invariant scaling limit for the height function of the Temperleyan dimer model on such graphs. Along the way, we describe the relationship between Temperleyan CRSFs and loop measures, and develop tools of independent interest to study the latter using only rough control on the random walk

Domaines: Probabilités [math.PR]

Nathanaël Berestycki, Benoit Laslier, Gourab Ray. Dimers on Riemann surfaces II: conformal invariance and scaling limit. 2022. ⟨hal-03733461⟩