Negative moments for Gaussian multiplicative chaos on fractal sets

@article{Garban2018NegativeMF,
  title={Negative moments for Gaussian multiplicative chaos on fractal sets},
  author={Christophe Garban and Nina Holden and Avelio Sep'ulveda and Xin Sun},
  journal={arXiv: Probability},
  year={2018}
}
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… 
View PDF on arXiv
5 Citations
Background Citations
2
Methods Citations
2

Figures from this paper

5 Citations

Liouville dynamical percolation
We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a $\gamma$-Liouville measure. The homogeneous counterpart of this process describes…
Volume of metric balls in Liouville quantum gravity
We study the volume of metric balls in Liouville quantum gravity (LQG). For $\gamma \in (0,2)$, it has been known since the early work of Kahane (1985) and Molchan (1996) that the LQG volume of…
The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity
We prove that the Tutte embeddings (a.k.a. harmonic/embeddings) of certain random planar maps converge to $\gamma$-Liouville quantum gravity ($\gamma$-LQG). Specifically, we treat mated-CRT maps,…
Path integral for quantum Mabuchi K-energy
We construct a path integral based on the coupling of the Liouville action and the Mabuchi K-energy on a one-dimensional complex manifold. To the best of our knowledge this is the first rigorous…
Local and global geometry of the 2D Ising interface in critical prewetting
Consider the Ising model at low temperatures and positive external field λ on an N×N box with Dobrushin boundary conditions that are plus on the north, east and west boundaries and minus on the south…

References

SHOWING 1-10 OF 30 REFERENCES
An elementary approach to Gaussian multiplicative chaos
A completely elementary and self-contained proof of convergence of Gaussian multiplicative chaos is given. The argument shows further that the limiting random measure is nontrivial in the entire…
Gaussian multiplicative chaos through the lens of the 2D Gaussian free field
The aim of this review-style paper is to provide a concise, self-contained and unified presentation of the construction and main properties of Gaussian multiplicative chaos (GMC) measures for…
Gaussian multiplicative chaos and applications: A review
In this article, we review the theory of Gaussian multiplicative chaos initially introduced by Kahane’s seminal work in 1985. Though this beautiful paper faded from memory until recently, it already…
Gaussian multiplicative chaos revisited
In this article, we extend the theory of multiplicative chaos for positive definite functions in Rd of the form f(x) = 2 ln+ T|x|+ g(x) where g is a continuous and bounded function. The construction…
The Fourier spectrum of critical percolation
Consider the indicator function f of a 2-dimensional percolation crossing event. In this paper, the Fourier transform of f is studied and sharp bounds are obtained for its lower tail in several…
On Gaussian multiplicative chaos
The tail expansion of Gaussian multiplicative chaos and the Liouville reflection coefficient
In this short note, we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d GFF on the unit disk with zero average on the unit circle. More specifically, we…
Equivalence of Liouville measure and Gaussian free field
Given an instance $h$ of the Gaussian free field on a planar domain $D$ and a constant $\gamma \in (0,2)$, one can use various regularization procedures to make sense of the Liouville quantum gravity…
Liouville dynamical percolation
We construct and analyze a continuum dynamical percolation process which evolves in a random environment given by a $\gamma$-Liouville measure. The homogeneous counterpart of this process describes…
The Fyodorov–Bouchaud formula and Liouville conformal field theory
  • G. Remy
  • Mathematics
    Duke Mathematical Journal
  • 2020
In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian…
...
...