Random Walks on Mated-CRT Planar Maps and Liouville Brownian Motion

Abstract

We prove a scaling limit result for random walk on certain random planar maps with its natural time parametrization. In particular, we show that for \(\gamma \in (0,2)\), the random walk on the mated-CRT map with parameter \(\gamma \) converges to \(\gamma \)-Liouville Brownian motion, the natural quantum time parametrization of Brownian motion on a \(\gamma \)-Liouville quantum gravity (LQG) surface. Our result applies if the mated-CRT map is embedded into the plane via the embedding which comes from SLE/LQG theory or via the Tutte embedding (a.k.a. the harmonic or barycentric embedding). In both cases, the convergence is with respect to the local uniform topology on curves and it holds in the quenched sense, i.e., the conditional law of the walk given the map converges. Previous work by Gwynne, Miller, and Sheffield (2017) showed that the random walk on the mated-CRT map converges to Brownian motion modulo time parametrization. This is the first work to show the convergence of the parametrized walk. As an intermediate result of independent interest, we derive an axiomatic characterisation of Liouville Brownian motion, for which the notion of Revuz measure of a Markov process plays a crucial role.