Massive SLE$_4$ and the scaling limit of the massive harmonic explorer

TL;DR

This paper rigorously proves that the scaling limit of the massive harmonic explorer converges to massive SLE$_4$, a conformally covariant curve, and establishes its coupling with a massive Gaussian free field as a level line.

Contribution

It provides a complete proof that the massive harmonic explorer converges to massive SLE$_4$ and demonstrates its coupling with a massive Gaussian free field.

Findings

The scaling limit of the massive harmonic explorer is a massive SLE$_4$.

Massive SLE$_4$ is conformally covariant and absolutely continuous with respect to SLE$_4$.

Massive SLE$_4$ can be coupled with a massive Gaussian free field as a level line.

Abstract

The massive harmonic explorer is a model of random discrete path on the hexagonal lattice that was proposed by Makarov and Smirnov as a massive perturbation of the harmonic explorer. They argued that the scaling limit of the massive harmonic explorer in a bounded domain is a massive version of chordal SLE$_4$, called massive SLE$_4$, which is conformally covariant and absolutely continuous with respect to chordal SLE$_4$. In this paper, we provide a full and rigorous proof of this statement. Moreover, we show that a massive SLE$_4$ curve can be coupled with a massive Gaussian free field as its level line, when the field has appropriate boundary conditions.