The double-dimer loop ensemble converges to CLE₄

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Resolves the half-plane Temperleyan form of the double-dimer scaling-limit conjecture: the complete loop ensemble formed by two independent dimer coverings of the Temperleyan square lattice converges to nested CLE₄. Convergence matches every macroscopic loop as an unparametrized curve, upgrading convergence of loop observables to convergence of the loops themselves.

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The paper states

Theorem 1.1

In The curve scaling limit of half-plane double dimers

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Proposition 1.3 (Standard properties of nested \CLE_4) in the same paper () and Theorem 1.2 (Cylindrical convergence [BI, Theorem 4.2(2)]) in the same paper ().

For the half-plane Temperleyan law specified above, the family F(\mathcal L_\delta) is tight as \delta\downarrow0 in the loop-collection topology just defined, and

F(\mathcal L_\delta)\ \Longrightarrow\ F(\mathcal L), \qquad \delta\downarrow0,

where \mathcal L is standard nested \CLE_4 in \HH. The convergence holds along the full mesh limit and matches all macroscopic loops as curves.

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The curve scaling limit of half-plane double dimers

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We prove that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE₄. The convergence holds along the full mesh limit and matches every macroscopic loop as an unparametrized curve. This resolves the half-plane Temperleyan form of the double-dimer CLE₄ scaling-limit conjecture.

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21 references in The curve scaling limit of half-plane double dimers

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