Electronic Communications in Probability

Negative moments for Gaussian multiplicative chaos on fractal sets

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

Full-text: Open access

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 $\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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1545188960

Digital Object Identifier
doi:10.1214/18-ECP168

Mathematical Reviews number (MathSciNet)
MR3896838

Zentralblatt MATH identifier
07023489

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

Keywords
Liouville measure log-correlated fields Gaussian free field negative moments

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

  • [Adl90] R. J. Adler. An introduction to continuity, extrema, and related topics for general Gaussian processes. IMS, 1990.
  • [Aru17] J. Aru. Gaussian multiplicative chaos through the lens of the 2D Gaussian free field. arXiv preprint, arXiv:1709.04355, 2017.
  • [Ber17] N. Berestycki. An elementary approach to Gaussian multiplicative chaos. Electronic communications in Probability, 22. 2017.
  • [BSS14] N. Berestycki, S. Sheffield, X. Sun. Equivalence of Liouville measure and Gaussian free field. arXiv preprint, arXiv:1410.5407, 2014.
  • [DS11] B. Duplantier, S. Sheffield. Liouville quantum gravity and KPZ. Inventiones mathematicae, 185, no. 2, 333–393, 2011.
  • [FB08] Y. V. Fyodorov and J-P. Bouchaud. Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A, 41(37):372001, 12, 2008.
  • [GH+18] C. Garban, N. Holden, A. Sepúlveda, X. Sun. Liouville dynamical percolation. In preparation. 2018.
  • [GPS10] C. Garban, G. Pete, O. Schramm. The Fourier spectrum of critical percolation. Acta Mathematica 205, no. 1, 19–104, 2010.
  • [Kah85] J-P Kahane. Sur le chaos multiplicatif. Annales des Sciences Mathématiques du Québec, 9(2):105–150, 1985.
  • [KP76] J-P. Kahane, J. Peyrière. Sur certaines martingales de Benoit Mandelbrot. Advances in mathematics 22, no. 2: 131–145, 1976.
  • [LRV18] H. Lacoin, R. Rhodes, V. Vargas. Path integral for quantum Mabuchi K-energy. arXiv preprint, arXiv:1807.01758, 2018.
  • [Pit82] L. D. Pitt. Positively correlated normal variables are associated. The Annals of Probability, 10(2):496–499, 1982.
  • [Rem17] G. Remy. The Fyodorov-Bouchaud formula and Liouville conformal field theory. arXiv preprint, arXiv:1710.06897, 2017.
  • [RV14] R. Rhodes, V. Vargas. Gaussian multiplicative chaos and applications: a review. Probability Surveys 11, 2014.
  • [RV16] R. Rhodes, V. Vargas. Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity. arXiv preprint, arXiv:1602.07323, 2016.
  • [RV17] R. Rhodes, V. Vargas. The tail expansion of Gaussian multiplicative chaos and the Liouville reflection coefficient. arXiv preprint, arXiv:1710.02096, 2017.
  • [RoV10] R. Robert, V. Vargas. Gaussian multiplicative chaos revisited. The Annals of Probability, 38, no. 2: 605–631, 2010.
  • [Sha16] A. Shamov. On Gaussian multiplicative chaos. Journal of Functional Analysis, 270(9):3224–3261, 2016.
  • [She07] S. Sheffield. Gaussian free fields for mathematicians. Probability theory and related fields, 139(3-4):521–541, 2007.