Conformally invariant fields out of Brownian loop soups

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.

arXiv comment80 pages, 4 figures. Improved presentation compared to first version