# Random-map geometry and diffusion research benchmark

Family 211: The geometric phase diagram, diffusion, and spectra of random planar maps. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Random-geometry researchers can compare FK planar-map phase and random-walk scaling limits with simulated observables.

## Applicability and commercial boundary

Continuum limits use model-specific Brownian/LQG inputs; they do not imply finite map routing or a geographic city diffusion model.

## Initial business decision

Research infrastructure. Buyer budget, commercial novelty and profitability are unvalidated.

## Next verification action

Extract a finite observable and the required scaling normalization.

## Evidence scope

The catalog statement was individually reviewed. This record does not imply a manuscript proof review or formal-scope comparison. No independent Lean check was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Main source

[Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026/fk-ising-walk-limit.pdf).