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

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