The double-dimer CLE_4 conjecture in the half-plane at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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The double-dimer $\mathrm{CLE}_4$ conjecture in the half-plane

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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If this game doesn't work on your computer, we can't help you. No Lean version yet. Some unformalized games could have issues!

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	luck, magnets	Levels:	1
Category:	Probability and statistical mechanics	Lean version:	not yet
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The double-dimer loop ensemble converges to CLE₄. 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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LEVEL 1 The curve scaling limit of half-plane double dimers

released 2026-09-23 | 2 theorems · 15 lemmas · 22 proofs · 17,693 words | PLAY LEVEL 1 » (pdf)

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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