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

TL;DR

This paper investigates the structure of level lines of the Gaussian Free Field (GFF) in the whole-plane setting, revealing a tree-structure of loops and a reversible exploration process starting from interior points.

Contribution

It introduces a novel tree-structure of level loops and a reversible continuum exploration process for whole-plane GFF, extending previous boundary-based results.

Findings

Level lines form a sequence of loops with a target-independent property.

The level loop sequences create a tree-structure of the plane.

The continuum exploration process of whole-plane GFF is reversible.

Abstract

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".