Pointed half-plane meandric-system scaling limit conjecture

Let (M,P,Γ)(\mathcal M^\circ,P^\circ,\Gamma^\circ) be the infinite random planar map decorated by a bi-infinite Hamiltonian path and loops and possibly infinite paths associated with the pointed infinite half-plane meandric system (PIHPMS). Let γ\gamma^\circ be the path started from 00 in Γ\Gamma^\circ, which is almost surely the unique infinite path. PIHPMS scaling limit conjecture. Under an appropriate scaling limit, (M,P,γ,Γγ)(\mathcal M^\circ,P^\circ,\gamma^\circ,\Gamma^\circ\setminus\gamma^\circ) converges to a 2\sqrt2-LQG wedge, a space-filling SLE8_8 from \infty to \infty in the half-plane, a chordal SLE6_6 from 00 to \infty in the half-plane, and a CLE6_6 in the complement of the SLE6_6 curve, equivalently a union of conditionally independent CLE6_6 ensembles in the complementary components. This continuum prediction concerns the pointed model and remains open.

Sources & referencesView supporting material

Primary source

Jacopo Borga, Ewain Gwynne and Minjae Park, “On the geometry of uniform meandric systems”, arXiv:2212.00534 (2023).

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