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${}_4$.

60 pages, 43 figures. This version adds a discussion about GFF with general boundary conditions at the end of the introduction and updates several references. arXiv admin note: text overlap with arXiv:1201.1496 by other authors

References in corpus (1)

• Level Lines of Gaussian Free Field II: Whole-Plane GFF

Cited by in corpus (4)

• Revisiting the combinatorics of the 2D Ising model
• Almost sure multifractal spectrum of SLE
• Level lines of the Gaussian Free Field with general boundary data
• Simple CLE in Doubly Connected Domains