Free-Boundary Dimer Convergence to Arc Loop Ensembles

Definitions and mathematical statement

Let $\mathbb H=\{w\in{\mathbb{C}}:\operatorname{Im}w>0\}$. Fix a monomer fugacity $z>0$. For mesh size $\delta>0$, write $\mathcal C_{\delta,z}$ for the collection of boundary-touching curves in the free-boundary dimer model approximating $\mathbb H$, with the model and curve construction intended by the supplied source. The record does not specify that construction or the discrete approximations.

The notation $\operatorname{ALE}_{-\lambda,\lambda}$ denotes the Arc Loop Ensemble with the source's boundary labels $-\lambda$ and $\lambda$, and $\operatorname{CLE}_4$ denotes the conformal loop ensemble with parameter $4$. The record does not define the ALE construction or the value and normalization of $\lambda$.

Conjecture (supplied formulation). For every fixed $z>0$, as $\delta\downarrow0$, \[ \mathcal C_{\delta,z}\ \xrightarrow{\mathrm{law}}\ \mathcal A, \qquad \mathcal A\sim\operatorname{ALE}_{-\lambda,\lambda}. \] In the limiting curve ensemble, conditional on $\mathcal A$, the loop collections $\mathcal L_D$, indexed by the connected components $D$ of $\mathbb H\setminus\mathcal A$, are mutually independent, and each $\mathcal L_D$ has the law of $\operatorname{CLE}_4$ in $D$.

This states the intended convergence target, but a fully specified convergence assertion requires the missing discrete model, construction of both boundary-touching curves and loops, ALE normalization, and topology of convergence. The conditional loop assertion is understood as part of the proposed limiting ensemble; the source record does not specify a separate discrete loop ensemble or its joint convergence with $\mathcal C_{\delta,z}$.

Short English statement

For every positive monomer fugacity, the conjecture predicts that the boundary-touching curves of the free-boundary dimer model in the upper half-plane approach $\operatorname{ALE}_{-\lambda,\lambda}$. Given this limiting ensemble, the remaining loop collections should be independent $\operatorname{CLE}_4$ ensembles in its complementary components. Several model and convergence conventions are absent from the supplied record and require mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004559$, title Free-Boundary Dimer Convergence to Arc Loop Ensembles, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied report citation: Mini-Workshop: Dimers, Ising and Spanning Trees beyond the Critical Isoradial Case (2021), Oberwolfach Reports, pp. 1759--1789, DOI 10.4171/owr/2020/35. https://doi.org/10.4171/owr/2020/35. The record describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Supplied literature source: Laslier and Ray, free-boundary dimers. https://arxiv.org/abs/2102.12873.
• Supplied literature source: Conformally invariant boundary arcs in double dimers. https://arxiv.org/abs/2409.18015.
• Supplied literature source: OWR 2020/35 report. https://doi.org/10.4171/owr/2020/35.
• The supplied literature assessment was checked on 2026-08-22. It labels the problem open and describes the full ALE and $\operatorname{CLE}_4$ convergence as conjectural, while reporting that recent work proves important boundary statistics. This is supplied dated curation derived from the OWR record, not an independently verified assessment; time-sensitive status and cited claims require independent verification.
• The registry assigns the statement status corrected_verified and reports that it was checked against the official Oberwolfach report for preservation of hypotheses, notation, and the open target. This verification claim is supplied metadata, not independent verification in this draft.