Liouville dynamical percolation

We construct and analyze a continuum dynamical percolation process which evolves in random environment given by $$\gamma $$ -Liouville measure. The homogeneous counterpart of this describes the scaling limit discrete on rescaled triangular lattice. Our focus here is to study same limiting dynamics, but where speed microscopic updates highly inhomogeneous space driven measure associated with two-dimensional log-correlated field h. Roughly speaking, very rapidly h high barely moves low. main results can be summarized as follows.