{"id":"openai-math/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps","pdf":"https://github.com/openai/math/blob/main/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/linear-clock-random-walk-tree-weighted-planar-maps.pdf","url":"https://github.com/openai/math/tree/main/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026","tags":["random-walk","planar-maps","probability"],"score":0,"title":"A Linear Clock for Random Walk on Tree-Weighted Planar Maps","domain":"Probability and statistical mechanics","family":"211","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.PR","summary":"We prove that stationary random walk on a planar map sampled with weight equal to its number of spanning trees converges to Liouville Brownian motion on the unit-area $`\\sqrt2`$-Liouville quantum sphere. The convergence retains the conditional path law jointly with the measured metric space. The walk chooses uniformly among all incident half-edges and starts from the stationary degree measure. For total attempt rate one and the continuum Dirichlet form with factor 1/2, the time acceleration is exactly the number of map edges. The result uses the companion contour and metric limits for this…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-05","reactions":{},"formalized":false,"family_title":"The geometric phase diagram, diffusion, and spectra of random planar maps"}