{"id":"openai-math/The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4","pdf":"https://github.com/openai/math/blob/main/preprints/The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026/main.pdf","url":"https://github.com/openai/math/tree/main/preprints/The-critical-Liouville-quantum-sphere-and-geometric-limits-of-FK-maps-at-q-equals-4-September-24-2026","tags":["probability","liouville-quantum-gravity","random-planar-maps","percolation"],"score":0,"title":"The critical Liouville quantum sphere and geometric limits of FK maps at q=4","domain":"Probability and statistical mechanics","family":"211","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We construct the field and area law of the unit-area critical Liouville quantum sphere as a limit of ordinary subcritical quantum spheres, and equip it with its critical intrinsic metric. We then prove that spherical Fortuin–Kasteleyn planar maps at q = 4, embedded by their equilateral flag uniformizations, converge jointly to this sphere decorated by an independent nested conformal loop ensemble CLE 4 . With deterministic distance normalization, the convergence includes the area measure, the full embedded distance function, and every macroscopic interface through all positive integer edge…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-24","reactions":{},"formalized":false,"family_title":"The geometric phase diagram, diffusion, and spectra of random planar maps"}