Electronic Communications in Probability

Negative moments for Gaussian multiplicative chaos on fractal sets

Christophe Garban, Nina Holden, Avelio Sepúlveda, and Xin Sun

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The objective of this note is to study the probability that the total mass of a subcritical Gaussian multiplicative chaos (GMC) with arbitrary base measure $\sigma $ is small. When $\sigma $ has some continuous density w.r.t Lebesgue measure, a scaling argument shows that the logarithm of the total GMC mass is sub-Gaussian near $-\infty $. However, when $\sigma $ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure $\sigma $, the total GMC mass has negative moments of all orders.

Article information

Electron. Commun. Probab., Volume 23 (2018), paper no. 100, 10 pp.

Received: 28 May 2018
Accepted: 11 September 2018
First available in Project Euclid: 19 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G60: Random fields 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]

Liouville measure log-correlated fields Gaussian free field negative moments

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Garban, Christophe; Holden, Nina; Sepúlveda, Avelio; Sun, Xin. Negative moments for Gaussian multiplicative chaos on fractal sets. Electron. Commun. Probab. 23 (2018), paper no. 100, 10 pp. doi:10.1214/18-ECP168. https://projecteuclid.org/euclid.ecp/1545188960

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