Level lines of Gaussian Free Field I: Zero-boundary GFF

We study level lines of Gaussian Free Field h emanating from boundary points. The article has two parts. In the first part, we show that the level lines are random continuous curves which are variants of SLE4 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 SLE4(ρ¯) process is still an SLE4(ρ¯) process. We prove that the level lines satisfy “target-independent” property. In the second part, we discuss the relation between Gaussian Free Field and Conformal Loop Ensemble (CLE). A CLE is a collection of disjoint SLE-loops. Since the level lines of GFF are SLE4 paths, the collection of level loops of GFF corresponds to CLE4. We study the coupling between GFF and CLE4 with time parameter which sheds lights on the conformally invariant metric on CLE4.