July 2022 Brownian loops and the central charge of a Liouville random surface
Morris Ang, Minjae Park, Joshua Pfeffer, Scott Sheffield
Author Affiliations +
Ann. Probab. 50(4): 1322-1358 (July 2022). DOI: 10.1214/21-AOP1558


We explore the geometric meaning of the so-called zeta-regularized determinant of the Laplace–Beltrami operator on a compact surface, with or without boundary. We relate the (c/2)th power of the determinant of the Laplacian to the appropriately regularized partition function of a Brownian loop soup of intensity c on the surface. This means that, in a certain sense, decorating a random surface by a Brownian loop soup of intensity c corresponds to weighting the law of the surface by the (c/2)th power of the determinant of the Laplacian.

Next, we introduce a method of regularizing a Liouville quantum gravity (LQG) surface (with some matter central charge parameter c) to produce a smooth surface. And we show that weighting the law of this random surface by the (c/2)th power of the Laplacian determinant has precisely the effect of changing the matter central charge from c to c+c. Taken together with the earlier results, this provides a way of interpreting an LQG surface of matter central charge c as a pure LQG surface decorated by a Brownian loop soup of intensity c.

Building on this idea, we present several open problems about random planar maps and their continuum analogs. Although the original construction of LQG is well defined only for c1, some of the constructions and questions also make sense when c>1.

Funding Statement

J.P. was partially supported by the National Science Foundation Graduate Research Fellowship Grant No. 1745302. M.A, M.P., and S.S. were partially supported by NSF Award DMS-1712862.


We thank Ewain Gwynne, Jason Miller, Peter Sarnak, Xin Sun, Yilin Wang, and Wendelin Werner for helpful conversations.


Download Citation

Morris Ang. Minjae Park. Joshua Pfeffer. Scott Sheffield. "Brownian loops and the central charge of a Liouville random surface." Ann. Probab. 50 (4) 1322 - 1358, July 2022. https://doi.org/10.1214/21-AOP1558


Received: 1 September 2020; Revised: 1 September 2021; Published: July 2022
First available in Project Euclid: 11 May 2022

MathSciNet: MR4420421
zbMATH: 1489.60014
Digital Object Identifier: 10.1214/21-AOP1558

Primary: 60D05

Keywords: Brownian loop soup , Laplacian , Liouville quantum gravity

Rights: Copyright © 2022 Institute of Mathematical Statistics


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Vol.50 • No. 4 • July 2022
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