Coupling the Gaussian Free Fields with Free and with Zero Boundary Conditions via Common Level Lines

Wendelin Werner,Wei Qian

doi:10.1007/s00220-018-3159-z

Abstract

We point out a new simple way to couple the Gaussian Free Field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample at random all the signs of the height gaps on its boundary-touching zero-level lines (these signs are alternating for the zero-boundary GFF) in order to obtain a free boundary GFF. Constructions and couplings of the free boundary GFF and its level lines via soups of reflected Brownian loops and their clusters are also discussed. Such considerations show for instance that in a domain with an axis of symmetry, if one looks at the overlay of a single usual Conformal Loop Ensemble CLE3 with its own symmetric image, one obtains the CLE4-type collection of level lines of a GFF with mixed zero/free boundary conditions in the half-domain.

Highlights

The ALE of a Dirichlet Gaussian Free Field (GFF)we will describe what we will refer to as the ALE1 of a GFF with Dirichlet boundary conditions
We point out a new simple way to couple the Gaussian Free Field (GFF) with free boundary conditions in a two-dimensional domain with the GFF with zero boundary conditions in the same domain: Starting from the latter, one just has to sample at random all the signs of the height gaps on its boundary-touching zero-level lines in order to obtain a free boundary GFF
Constructions and couplings of the free boundary GFF and its level lines via soups of reflected Brownian loops and their clusters are discussed. Such considerations show for instance that in a domain with an axis of symmetry, if one looks at the overlay of a single usual Conformal Loop Ensemble CLE3 with its own symmetric image, one obtains the CLE4-type collection of level lines of a GFF with mixed zero/free boundary conditions in the half-domain

Summary

Background

We will describe what we will refer to as the ALE1 of a GFF with Dirichlet boundary conditions. We will use the same procedure when we will consider and define the Gaussian Free Fields with different boundary conditions (Neumann, or mixed Neumann–Dirichlet) in a domain D as random processes indexed by the set of continuous compactly supported functions in D. One can couple a GFF with Dirichlet boundary conditions with an ALE as follows: Toss first one fair coin ε ∈ {−1, 1} in order to decide the label of the connected component of O that contains the origin, and define deterministically the labels ε j of all the other components O j of O in such a way that any two adjacent components have opposite labels Note that the number of iterations needed before discovering the CLE4 loop that surrounds a given point is random (which explains why the dimension of the CLE4 carpet is larger than 3/2)

The Coupling of the Two Fields

Proof of the Coupling

Relation to Soups of Reflected Loops and Consequences

Shifting the Scalar Version of the Neumann GFF and Related Comments

Further Questions and Remarks