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Liouville quantum gravity weighted by conformal loop ensemble nesting statistics

Nina Holden and Matthis Lehmkuehler

Vol. 5 (2024), No. 4, 847–960
DOI: 10.2140/pmp.2024.5.847
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 0 marked points z1,,zn. The idea is to consider a reweighting by B{1,,n}eσBNB, where σB and NB is the number of CLE loops surrounding the points zi for i 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 z1,,zn with a magnitude depending explicitly on σ1,,σn. We define the partition function of the surface, compute it for n {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 (σB : B {1,,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.

Keywords
conformal loop ensemble, Schramm–Loewner evolution, Liouville quantum gravity
Mathematical Subject Classification
Primary: 60G57, 60J67, 60J80, 60K37, 81T40
Secondary: 05C80, 60G52, 60G60, 60K05, 60K35
Milestones
Received: 4 May 2022
Revised: 27 March 2024
Accepted: 11 June 2024
Published: 18 July 2024
Authors
Nina Holden
Courant Institute
New York University
New York, NY
United States
Matthis Lehmkuehler
Departement Mathematik und Informatik
Universität Basel
Basel
Switzerland