Conformally invariant fields out of Brownian loop soups

TL;DR

This paper constructs and analyzes a conformally invariant field $h_\theta$ associated with Brownian loop soups in 2D, generalizing the Gaussian free field and establishing connections with conformal loop ensembles and Schramm-Loewner evolutions.

Contribution

It introduces a new family of conformally invariant fields $h_\theta$ for $\theta \in (0,1/2]$, extending the GFF case and linking loop soup clusters to Minkowski content and level lines.

Findings

For $\theta=1/2$, $h_\theta$ recovers the Gaussian free field.

The field $h_\theta$ can be obtained via fractional derivatives of a multiplicative chaos measure.

CLE$_\kappa$ loops serve as level lines for $h_\theta$ with a constant height gap.

Abstract

Consider a Brownian loop soup $\mathcal{L}_D^\theta$ with subcritical intensity $\theta \in (0,1/2]$ in some 2D bounded simply connected domain. We define and study the properties of a conformally invariant field $h_\theta$ naturally associated to $\mathcal{L}_D^\theta$. Informally, this field is a signed version of the local time of $\mathcal{L}_D^\theta$ to the power $1-\theta$. When $\theta=1/2$, $h_\theta$ is a Gaussian free field (GFF) in $D$. Our construction of $h_\theta$ relies on the multiplicative chaos $\mathcal{M}_\gamma$ associated with $\mathcal{L}_D^\theta$, as introduced in [ABJL23]. Assigning independent symmetric signs to each cluster, we restrict $\mathcal{M}_\gamma$ to positive clusters. We prove that, when $\theta=1/2$, the resulting measure $\mathcal{M}_\gamma^+$ corresponds to the exponential of $\gamma$ times a GFF. At this intensity, the GFF can be recovered by differentiating at $\gamma=0$ the measure $\mathcal{M}_\gamma^+$. When $\theta<1/2$, we show that $\mathcal{M}_\gamma^+$ has a nondegenerate fractional derivative at $\gamma=0$ defining a random generalised function $h_\theta$. We establish a result which is analoguous to the recent work [ALS23] in the GFF case ($\theta=1/2$), but for $h_\theta$ with $\theta \in (0,1/2]$. Relying on the companion article [JLQ23], we prove that each cluster of $\mathcal{L}_D^\theta$ possesses a nondegenerate Minkowski content in some non-explicit gauge function $r \mapsto r^2 |\log r|^{1-\theta+o(1)}$. We then prove that $h_\theta$ agrees a.s. with the sum of the Minkowski content of each cluster multiplied by its sign. We further extend the couplings between CLE$_4$, SLE$_4$ and the GFF to $h_\theta$ for $\theta\in(0,1/2]$. We show that the (non-nested) CLE$_\kappa$ loops form level lines for $h_\theta$ and that there exists a constant height gap between the values of the field on either side of the CLE loops.