# Coupling the Gaussian free fields with free and with zero boundary conditions via common level lines

## TL;DR

This paper introduces a simple coupling method for Gaussian Free Fields with free and zero boundary conditions using boundary level line sign sampling, and discusses related loop and cluster constructions.

## Contribution

It presents a novel coupling technique for GFFs with different boundary conditions via boundary level line sign sampling and explores their loop and cluster representations.

## Key findings

- Coupling free and zero boundary GFFs through boundary sign sampling.
- Connection between CLE(3) and CLE(4) via symmetric overlays of loop ensembles.
- Insights into level line structures of GFFs with mixed boundary conditions.

## 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 CLE(3) with its own symmetric image, one obtains the CLE(4)-type collection of level lines of a GFF with mixed zero/free boundary conditions in the half-domain.