Liouville quantum gravity weighted by conformal loop ensemble nesting statistics

Abstract

We study Liouville quantum gravity (LQG) surfaces whose law has been reweighted according to nesting statistics for a conformal loop ensemble (CLE) relative to $n\in {\mathbb{N}}_0$ marked points $z_1,\dots ,z_n$. The idea is to consider a reweighting by ${\prod }_{B\subseteq \{1,\dots ,n\}}{e}^{{\sigma }_B{N}_B}$, where ${\sigma }_B\in \mathbb{R}$ and $N_B$ is the number of CLE loops surrounding the points $z_i$ for $i\in B$. This is made precise via an approximation procedure where as part of the proof we derive strong spatial independence results for CLE. The reweighting induces logarithmic singularities for the Liouville field at $z_1,\dots ,z_n$ with a magnitude depending explicitly on ${\sigma }_1,\dots ,{\sigma }_n$. We define the partition function of the surface, compute it for $n\in \{0,1\}$, and derive a recursive formula expressing the $n>1$ point partition function in terms of lower-order partition functions. The proof of the latter result is based on a continuum peeling process previously studied by Miller, Sheffield and Werner in the case $n=0$, and we derive an explicit formula for the generator of a boundary length process that can be associated with the exploration for general $n$. We use the recursive formula to partly characterize for which values of $({\sigma }_B:B\subseteq \{1,\dots ,n\})$ the partition function is finite. Finally, we give a new proof for the law of the conformal radius of CLE, which was originally established by Schramm, Sheffield, and Wilson.

96 pages, 9 figures

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