Liouville dynamical percolation

Abstract

We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a $\gamma$-Liouville measure. The homogeneous counterpart of this process describes the scaling limit of discrete dynamical percolation on the rescaled triangular lattice. Our focus here is to study the same limiting dynamics, but where the speed of microscopic updates is highly inhomogeneous in space and is driven by the $\gamma$-Liouville measure associated with a two-dimensional log-correlated field $h$. Roughly speaking, this continuum percolation process evolves very rapidly where the field $h$ is high and barely moves where the field $h$ is low. Our main results can be summarized as follows. 1. First, we build this inhomogeneous dynamical percolation which we call $\gamma$-Liouville dynamical percolation (LDP) by taking the scaling limit of the associated process on the triangular lattice. We work with three different regimes each requiring different tools: $\gamma\in [0,2-\sqrt{5/2})$, $\gamma\in [2-\sqrt{5/2}, \sqrt{3/2})$, and $\gamma\in(\sqrt{3/2},2)$. 2. When $\gamma<\sqrt{3/2}$, we prove that $\gamma$-LDP is mixing in the Schramm-Smirnov space as $t\to \infty$, quenched in the log-correlated field $h$. On the contrary, when $\gamma>\sqrt{3/2}$ the process is frozen in time. The ergodicity result is a crucial piece of the Cardy embedding project of the second and fourth coauthors, where LDP for $\gamma=\sqrt{1/6}$ is used to study the scaling limit of a variant of dynamical percolation on uniform triangulations. 3. When $\gamma<\sqrt{3/4}$, we obtain quantitative bounds on the mixing of quad crossing events.