# First passage sets of the 2D continuum Gaussian free field

## TL;DR

This paper introduces the first passage set (FPS) for the 2D continuum Gaussian free field, characterizing its properties, construction, and fractal nature, and links it to Gaussian multiplicative chaos theory.

## Contribution

It provides the first axiomatic and constructive description of the FPS for the 2D GFF, including its fractal properties and measure-theoretic characterization.

## Key findings

- FPS is a fractal set with zero Lebesgue measure and Minkowski dimension 2.
- The associated measure is identified as a Minkowski content measure in a specific non-integer gauge.
- The FPS is coupled with the GFF as a local set, with a measure related to Gaussian multiplicative chaos.

## Abstract

We introduce the first passage set (FPS) of constant level $-a$ of the two-dimensional continuum Gaussian free field (GFF) on finitely connected domains. Informally, it is the set of points in the domain that can be connected to the boundary by a path on which the GFF does not go below $-a$. It is, thus, the two-dimensional analogue of the first hitting time of $-a$ by a one-dimensional Brownian motion. We provide an axiomatic characterization of the FPS, a continuum construction using level lines, and study its properties: it is a fractal set of zero Lebesgue measure and Minkowski dimension 2 that is coupled with the GFF $\Phi$ as a local set $A$ so that $\Phi+a$ restricted to $A$ is a positive measure. One of the highlights of this paper is identifying this measure as a Minkowski content measure in the non-integer gauge $r \mapsto \vert\log(r)\vert^{1/2}r^{2}$, by using Gaussian multiplicative chaos theory.