A Linear Clock for Random Walk on Tree-Weighted Planar Maps

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 $\sqrt{2}$-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 same ensemble.