Electronic Communications in Probability

Critical Liouville measure as a limit of subcritical measures

Juhan Aru, Ellen Powell, and Avelio Sepúlveda

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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

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

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

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Mathematical Reviews number (MathSciNet)

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)

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

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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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