Critical percolation phase-transition conjecture for supercritical random planar maps

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Let Q∈(0,2)Q\in(0,2) and let pc(Q)∈(0,1)p_c(Q)\in(0,1) be the critical probability from the non-trivial percolation transition conjecture. Consider critical site percolation on Mk\mathcal M_k with open boundary condition. Critical percolation connectivity conjecture. If Q>91/24Q>\sqrt{91/24}, equivalently c>95/4{\mathbf{c}}>95/4, then almost surely there is no infinite open cluster. If Q<91/24Q<\sqrt{91/24}, equivalently c<95/4{\mathbf{c}}<95/4, then with positive probability the open cluster containing ∂Mk\partial\mathcal M_k is infinite and has uncountably many ends. This conjecture is motivated by the expected CLE6_6 scaling limit and the Hausdorff dimension 91/4891/48 of the CLE6_6 gasket; the critical case Q=91/24Q=\sqrt{91/24} is not specified.

References

Primary source

Ewain Gwynne and Jinwoo Sung, “Random walk reflected off of infinity, with applications to uniform spanning forests and supercritical Liouville quantum gravity”, arXiv:2506.18827 (2026).

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