Conformally invariant fields out of Brownian loop soups

Abstract

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