An invariance principle for ergodic scale-free random environments

TL;DR

This paper proves that random walks in certain scale-invariant ergodic planar environments converge to Brownian motion, extending classical results to environments with variable local scales, relevant in quantum gravity and random map studies.

Contribution

It introduces a scale-invariance modulo scaling condition and proves quenched invariance principles for random walks in these environments, generalizing previous ergodic results.

Findings

Random walk convergence to Brownian motion in scale-invariant environments

Environment conditions include translation invariance modulo scaling and finite energy

Applications to random planar maps and Liouville quantum gravity

Abstract

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 {\em modulo 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. Our main result is that under modest constraints (translation invariance modulo scaling together with the finiteness of a type of specific energy) a random walk in this kind of environment converges to Brownian motion modulo time parameterization in the quenched sense. Environments of the type considered here arise naturally in the study of random planar maps and Liouville quantum gravity. In fact, the results of this paper are used in separate works to prove that certain random planar maps (embedded in the plane via the so-called Tutte embedding) have scaling limits given by SLE-decorated Liouville quantum gravity, and also to provide a more explicit construction of Brownian motion on the Brownian map. However, the results of this paper are much more general and can be read independently of that program. One general consequence of our main result is that if a translation invariant (modulo scaling) random embedded planar map and its dual have finite energy per area, then they are close on large scales to a minimal energy embedding (the harmonic embedding). To establish Brownian motion convergence for an {\em infinite} energy embedding, it suffices to show that one can perturb it to make the energy finite.