For every non-frozen translation-invariant ergodic Gibbs measure of the loop O(1) model at x=8 on the hexagonal lattice, the complement to the dimer configuration has either infinitely many loops around every face or a unique bi-infinite path.
Percolation and $O(1)$ loop model
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abstract
We present an "ultimate" proof of Cardy's formula for the critical percolation on the hexagonal lattice \cite{Smirnov01criticalpercolation}, showing the existence of the universal and conformally invariant scaling limit of crossing probabilities. The new approach is more conceptual, less technically demanding, and is amenable to generalizations.
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On loops in the complement to dimers
For every non-frozen translation-invariant ergodic Gibbs measure of the loop O(1) model at x=8 on the hexagonal lattice, the complement to the dimer configuration has either infinitely many loops around every face or a unique bi-infinite path.