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Conformally invariant fields out of Brownian loop soups by Jego, Antoine; Lupu, Titus; Qian, Wei is a scholarly article available to read on EtoBox.

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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 t

Author

Jego, Antoine; Lupu, Titus; Qian, Wei

Published

2023

Language

EN