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

Menglu Wang, Hao Wu

Published 2015-04-17Version 1

We study the level lines of GFF starting from interior points. We show that the level line of GFF starting from an interior point turns out to be a sequence of level loops. The sequence of level loops satisfies "target-independent" property. All sequences of level loops starting from interior points give a tree-structure of the plane. We also introduce the continuum exploration process of GFF starting from interior. The continuum exploration process of whole-plane GFF satisfies "reversibility". Finally, we explain the relation between whole-plane GFF and whole-plane CLE$_4$ and derive the fact that whole-plane CLE$_4$ is conformal invariant under any M\"obius transformation of the Riemann sphere from the reversibility of the continuum exploration process of whole-plane GFF.