Negative moments for gaussian multiplicative chaos on fractal sets

Christophe Garban; Nina Holden; Avelio Sepúlveda; Xin Sun

doi:10.1214/18-ECP168

Negative moments for gaussian multiplicative chaos on fractal sets

Christophe Garban, Nina Holden, Avelio Sepúlveda, Xin Sun

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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 σ is small. When σ 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 −∞. However, when σ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure σ, the total GMC mass has negative moments of all orders.

Original language	English (US)
Pages (from-to)	1-10
Number of pages	10
Journal	Electronic Communications in Probability
Volume	23
DOIs	https://doi.org/10.1214/18-ECP168
State	Published - 2018

Keywords

Gaussian free field
Liouville measure
Log-correlated fields
Negative moments

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

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10.1214/18-ECP168

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AU - Holden, Nina

AU - Sepúlveda, Avelio

AU - Sun, Xin

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N2 - 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 σ is small. When σ 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 −∞. However, when σ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure σ, the total GMC mass has negative moments of all orders.

AB - 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 σ is small. When σ 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 −∞. However, when σ has no scaling properties, the situation is much less clear. In this paper, we prove that for any base measure σ, the total GMC mass has negative moments of all orders.

KW - Gaussian free field

KW - Liouville measure

KW - Log-correlated fields

KW - Negative moments

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JF - Electronic Communications in Probability

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Negative moments for gaussian multiplicative chaos on fractal sets

Abstract

Keywords

ASJC Scopus subject areas

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