Scaling limit of critical FK planar maps for q>4
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For fixed q>4, what is the scaling limit of critical finite Fortuin–Kasteleyn planar maps with n edges? The target scales graph distances by a q-dependent constant times n^(−1/2) and uses normalized degree measure, with Brownian continuum random tree convergence in Gromov–Hausdorff–Prokhorov topology.
Why it matters and what counts as progress
Why it matters: The tree limit tests a predicted change of geometry in the q>4 finite-map regime.
What counts as progress: Prove the tree limit for the intended finite-map law or quantify the scaling constants and errors.
How the problem developed
- 10 Nov 2023 · Partial result [1]
Yuyang Feng Proved an infinite-volume local Gromov–Hausdorff–Prokhorov scaling limit to a continuum random tree for the q>4 critical FK planar-map model.
- 24 Sept 2026 · Proof claim [2]
OpenAI A manuscript claims convergence of finite critical FK planar maps for each fixed q>4 to the Brownian continuum random tree in Gromov–Hausdorff–Prokhorov topology after q-dependent n⁻¹ᐟ² distance scaling.
AI contribution
Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four
For each fixed q>4, critical finite Fortuin–Kasteleyn planar maps with n edges, distances scaled by a q-dependent constant times n⁻¹ᐟ², and normalized degree measure converge in distribution to the Brownian continuum random tree in Gromov–Hausdorff–Prokhorov topology.
Sources
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Publication dates and editorial changes.
AI-reported continuum tree limit for critical FK planar maps