The double-dimer nesting field converges to the CLE(4) nesting field in Sobolev spaces as the mesh size tends to zero, and the local loop count satisfies a central limit theorem.
Dimers on Riemann surfaces II: conformal invariance and scaling limit
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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
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Nesting of double-dimer loops: local fluctuations and convergence to the nesting field of CLE(4)
The double-dimer nesting field converges to the CLE(4) nesting field in Sobolev spaces as the mesh size tends to zero, and the local loop count satisfies a central limit theorem.