Liouville dynamical percolation

Understand

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.