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arXiv · 1805.00864

Negative moments for Gaussian multiplicative chaos on fractal sets

Christophe Garban·Nina Holden·Avelio Sepúlveda·Xin Sun
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Abstract

The objective of this note is to study the probability that the total mass of a sub-critical 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.

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Christophe Garban, Nina Holden, Avelio Sepúlveda, Xin Sun. 2018-05-02. Negative moments for Gaussian multiplicative chaos on fractal sets. https://arxiv.org/abs/1805.00864

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