An invariance principle for ergodic scale-free random environments

There are many classical random walk in environment results that apply to ergodic planar environments. We extend some of these environments which the length scale varies from place place, so law is a certain sense only translation invariant {\em modulo scaling}. For our purposes, an ``environment'' consists infinite map embedded $\mathbb C$, each whose edges comes with positive real conductance. Our main result under modest constraints (translation invariance scaling together finiteness type specific energy) this kind converges Brownian motion time parameterization quenched sense. Environments considered here arise naturally study maps and Liouville quantum gravity. In fact, paper used separate works prove (embedded plane via so-called Tutte embedding) have limits given by SLE-decorated gravity, also provide more explicit construction on map. However, much general can be read independently program. One consequence if (modulo scaling) its dual finite energy per area, then they close large scales minimal embedding (the harmonic embedding). To establish convergence for infinite} embedding, it suffices show one perturb make finite.