An invariance principle for random walk in ergodic scale-free random environments
There are many classical random walk in random environment results that apply to ergodic random planar environments. We extend some of these results to random environments in which the length scale varies from place to place, so that the law of the environment is in a certain sense only translation invariant modulo spatial scaling. For our purposes, an “environment” consists of an infinite random planar map embedded in $mathbb C$, each of whose edges comes with a positive real conductance. We show that under modest constraints (translation invariance modulo scaling together with a certain finite expectation hypothesis) a random walk in this kind of environment converges to Brownian motion modulo time parameterization in the quenched sense.
This result contains a number of existing scaling limit results for random walk in planar random environments. It also has important applications to random planar maps and Liouville quantum gravity (LQG). In particular, using our result we show that the Tutte embeddings of certain random planar maps (the-so-called mated-CRT maps) converge to LQG, which constitutes the first such scaling limit result for embedded random planar maps. We also give a direct construction of Brownian motion on the Brownian map as a limit of explicit discrete approximations.
Based on joint work with Jason Miller and Scott Sheffield.