Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Thick points for Gaussian free fields with different cut-offs

Alessandra Cipriani and Rajat Subhra Hazra

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Massive and massless Gaussian free fields can be described as generalized Gaussian processes indexed by an appropriate space of functions. In this article we study various approaches to approximate these fields and look at the fractal properties of the thick points of their cut-offs. Under some sufficient conditions for a centered Gaussian process with logarithmic variance we study the set of thick points and derive their Hausdorff dimension. We prove that various cut-offs for Gaussian free fields satisfy these assumptions. We also give sufficient conditions for comparing thick points of different cut-offs.


Les champs libres gaussiens massifs et sans masse peuvent être décrits comme des processus gaussiens généralisées indexés par un espace fonctionnel approprié. Dans cet article nous abordons différentes approches pour approximer ces champs et nous considérons les propriétés fractales des points épais de leur cut-off. Sous certaines conditions suffisantes, pour un processus gaussien avec variance logarithmique nous étudions l’ensemble des points épais et obtenons leur dimension de Hausdorff. Nous prouvons que différents cut-off des champs libres gaussiens satisfont ces hypothèses. Nous donnons aussi des conditions suffisantes pour comparer les points épais des différents cut-off.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 79-97.

Received: 5 August 2014
Revised: 14 August 2015
Accepted: 15 August 2015
First available in Project Euclid: 8 February 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G60: Random fields
Secondary: 60G15: Gaussian processes

Gaussian multiplicative chaos Cut-offs Liouville quantum gravity Thick points Hausdorff dimension


Cipriani, Alessandra; Hazra, Rajat Subhra. Thick points for Gaussian free fields with different cut-offs. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 79--97. doi:10.1214/15-AIHP709.

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  • [1] R. Adler. An Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes. IMS Lecture Notes – Monograph Series. Institute of Mathematical Statistics, Hayward, CA, 1990. Available at
  • [2] P. Balança and E. Herbin. A set-indexed Ornstein–Uhlenbeck process. Electron. Commun. Probab. 17 (2012) Article ID 39.
  • [3] S. Chatterjee. Chaos, concentration, and multiple valleys. Preprint, 2008. Available at arXiv:0810.4221.
  • [4] J. Dubédat. SLE and the free field: Partition functions and couplings. J. Amer. Math. Soc. 22 (4) (2009) 995–1054.
  • [5] B. Duplantier, R. Rhodes, S. Sheffield and V. Vargas. Renormalization of critical Gaussian multiplicative chaos and KPZ relation. Comm. Math. Phys. 330 (1) (2014) 283–330.
  • [6] B. Duplantier and S. Sheffield. Liouville quantum gravity and KPZ. Invent. Math. 185 (2) (2011) 333–393.
  • [7] C. Garban, R. Rhodes and V. Vargas. Liouville Brownian motion. Ann. Probab. 44 (4) (2016) 3076–3110.
  • [8] I. M. Gel’fand and N. Y. Vilenkin. Generalized Functions, Vol. 4: Applications of Harmonic Analysis. Academic Press, New York, 1964. Translated by Amiel Feinstein.
  • [9] T. Hida. White Noise: An Infinite Dimensional Calculus. Mathematics and Its Applications. Springer, Berlin, 1993. Available at
  • [10] J. Hu and M. Zähle. Generalized Bessel and Riesz potentials on metric measure spaces. Potential Anal. 30 (4) (2009) 315–340.
  • [11] X. Hu, J. Miller and Y. Peres. Thick points of the Gaussian free field. Ann. Probab. 38 (2) (2010) 896–926.
  • [12] J.-P. Kahane. Sur le chaos multiplicatif. Ann. Sci. Math. Québec 9 (2) (1985) 105–150.
  • [13] G. Lawler. Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. Available at
  • [14] M. Lifshits. Lectures on Gaussian Processes. Springer Briefs in Mathematics. Springer, Heidelberg, 2012.
  • [15] P. Mörters, Y. Peres, O. Schramm and W. Werner. Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. Available at
  • [16] R. Rhodes and V. Vargas. Gaussian multiplicative chaos and applications: A review. Probab. Surv. 11 (2014) 315–392.
  • [17] R. Rhodes and V. Vargas. Lectures on Gaussian multiplicative chaos. Random geometry program. University of Cambridge, Cambridge, 2015. Available at
  • [18] R. Robert and V. Vargas. Gaussian multiplicative chaos revisited. Ann. Probab. 38 (2) (2010) 605–631.
  • [19] L. Saloff-Coste. The heat kernel and its estimates. In Probabilistic Approach to Geometry. Proceedings of the 1st International Conference 405–436. M. Kotani. (Eds). Advanced Studies in Pure Mathematics 57. Mathematical Society of Japan, Tokyo, 2010.
  • [20] A. Shamov. On Gaussian multiplicative chaos. Preprint, 2014. Available at arXiv:1407.4418.
  • [21] S. Sheffield. Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 (3–4) (2007) 521–541.
  • [22] E. M. Stein. Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30. Princeton University Press, Princeton, NJ, 1970.
  • [23] A. Yaglom. Some classes of random fields in $n$-dimensional space, related to stationary random processes. Theory Probab. Appl. 2 (3) (1957) 273–320.