Dimers on Riemann surfaces II: conformal invariance and scaling limit

TL;DR

This paper establishes the existence of a universal, conformally invariant scaling limit for the Temperleyan cycle-rooted spanning forest on Riemann surfaces, advancing understanding of conformal invariance in statistical physics models.

Contribution

It proves the conformal invariance and universality of the scaling limit for CRSFs on Riemann surfaces, extending previous results and developing new tools for loop measure analysis.

Findings

Existence of conformally invariant scaling limit for CRSFs

Connection between CRSFs and loop measures

Development of new analytical tools for random walk analysis

Abstract

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