Level lines of the Gaussian Free Field with general boundary data

TL;DR

This paper investigates the existence and properties of level lines of the Gaussian free field with general boundary data, extending the SLE$_4$ process to include a continuum of force points and establishing a key monotonicity property.

Contribution

It introduces a framework for defining and analyzing level lines of the GFF with regulated boundary data, generalizing SLE$_4( ho)$ processes with a continuum of force points.

Findings

Level lines exist as continuous curves under certain boundary conditions.

Level lines are almost surely determined by the Gaussian free field.

A new monotonicity property related to boundary data is established.

Abstract

We study the level lines of a Gaussian free field in a planar domain with general boundary data $F$. We show that the level lines exist as continuous curves under the assumption that $F$ is regulated (i.e., admits left and right limits at every point), and satisfies certain inequalities. Moreover, these level lines are a.s. determined by the field. This allows us to define and study a generalization of the SLE$_4(\underline{\rho})$ process, now with a continuum of force points. A crucial ingredient is a monotonicity property in terms of the boundary data which strengthens a result of Miller and Sheffield and is also of independent interest.