Electronic Communications in Probability

Critical Liouville measure as a limit of subcritical measures

Juhan Aru, Ellen Powell, and Avelio Sepúlveda

Full-text: Open access

Abstract

We study how the Gaussian multiplicative chaos (GMC) measures $\mu ^\gamma $ corresponding to the 2D Gaussian free field change when $\gamma $ approaches the critical parameter $2$. In particular, we show that as $\gamma \to 2^{-}$, $(2-\gamma )^{-1}\mu ^\gamma $ converges in probability to $2\mu '$, where $\mu '$ is the critical GMC measure.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 18, 16 pp.

Dates
Received: 9 March 2018
Accepted: 5 January 2019
First available in Project Euclid: 23 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1553306557

Digital Object Identifier
doi:10.1214/19-ECP209

Mathematical Reviews number (MathSciNet)
MR3933042

Subjects
Primary: 60G15: Gaussian processes 60G60: Random fields 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G60: Random fields 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J67: Stochastic (Schramm-)Loewner evolution (SLE)

Keywords
Multiplicative chaos Liouville measure Liouville quantum gravity multiplicative cascades critical GMC measure first passage sets Gaussian free field local sets

Rights
Creative Commons Attribution 4.0 International License.

Citation

Aru, Juhan; Powell, Ellen; Sepúlveda, Avelio. Critical Liouville measure as a limit of subcritical measures. Electron. Commun. Probab. 24 (2019), paper no. 18, 16 pp. doi:10.1214/19-ECP209. https://projecteuclid.org/euclid.ecp/1553306557


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References

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