Level Lines of Gaussian Free Field I: Zero-Boundary GFF

Abstract

Let $h$ be an instance of Gaussian Free Field in a planar domain. We study level lines of $h$ starting from boundary points. We show that the level lines are random continuous curves which are variants of SLE$_4$ path. We show that the level lines with different heights satisfy the same monotonicity behavior as the level lines of smooth functions. We prove that the time-reversal of the level line coincides with the level line of $-h$. This implies that the time-reversal of SLE$_4(\underline{\rho})$ process is still an SLE$_4(\underline{\rho})$ process. We prove that the level lines satisfy "target-independent" property. We also discuss the relation between Gaussian Free Field and CLE$_4$.