Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

The sharp phase transition for level set percolation of smooth planar Gaussian fields

Stephen Muirhead and Hugo Vanneuville

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove that the connectivity of the level sets of a wide class of smooth centred planar Gaussian fields exhibits a phase transition at the zero level that is analogous to the phase transition in Bernoulli percolation. In addition to symmetry, positivity and regularity conditions, we assume only that correlations decay polynomially with exponent larger than two – roughly equivalent to the integrability of the covariance kernel – whereas previously the phase transition was only known in the case of the Bargmann–Fock covariance kernel which decays super-exponentially. We also prove that the phase transition is sharp, demonstrating, without any further assumption on the decay of correlations, that in the sub-critical regime crossing probabilities decay exponentially.

Key to our methods is the white-noise representation of a Gaussian field; we use this on the one hand to prove new quasi-independence results, inspired by the notion of influence from Boolean functions, and on the other hand to establish sharp thresholds via the OSSS inequality for i.i.d. random variables, following the recent approach of Duminil-Copin, Raoufi and Tassion.


Nous démontrons l’existence d’un phénomène de transition de phase pour les propriétés de connexion de lignes de niveau d’une grande classe de champs gaussiens planaires. En plus d’hypothèses de symétrie, régularié et positivité des corrélations, nous supposons que la covariance de ces champs décroît à vitesse polynomiale avec un exposant strictement plus grand que $2$. Nous montrons par ailleurs que la transition de phase est “nette” dans le sens que, dans la phase sous-critique, les probabilités de connexion convergent vers $0$ à vitesse exponentielle. Dans nos preuves, nous utilisons de façon centrale l’écriture des champs gaussiens lisses à l’aide d’un bruit blanc planaire. Nous utilisons cette représentation pour prouver de nouveaux résultats de quasi-indépendance spatiale, inspirés par la notion d’influence en théorie des fonctions booléennes. Par ailleurs, la structure produit du buit blanc nous permet d’utiliser l’inégalité d’OSSS, qui est une inégalité clef dans la récente approche d’étude des transitions de phase par Duminil-Copin, Raoufi et Tassion.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 56, Number 2 (2020), 1358-1390.

Received: 25 January 2019
Accepted: 27 May 2019
First available in Project Euclid: 16 March 2020

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 60G60: Random fields 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Gaussian fields Percolation Sharp phase transition


Muirhead, Stephen; Vanneuville, Hugo. The sharp phase transition for level set percolation of smooth planar Gaussian fields. Ann. Inst. H. Poincaré Probab. Statist. 56 (2020), no. 2, 1358--1390. doi:10.1214/19-AIHP1006.

Export citation


  • [1] R. Adler and J. Taylor. Random Fields and Geometry. Springer, New York, 2007.
  • [2] D. Ahlberg and R. Baldasso. Noise sensitivity and Voronoi percolation. Electron. J. Probab. 23 (2018) 108.
  • [3] K. Alexander. Boundedness of level lines for two-dimensional random fields. Ann. Probab. 24 (4) (1996) 1653–1674.
  • [4] J. Azaïs and M. Wschebor. Level Sets and Extrema of Random Processes and Fields. John Wiley & Sons, Inc., Hoboken, NJ, 2009.
  • [5] V. Beffara and D. Gayet. Percolation of random nodal lines. Publ. Math. 126 (2017) 131–176.
  • [6] D. Beliaev and S. Muirhead. Discretisation schemes for level sets of planar Gaussian fields. Comm. Math. Phys. 359 (2018) 869–913.
  • [7] D. Beliaev, S. Muirhead and I. Wigman. Russo–Seymour–Welsh estimates for the Kostlan ensemble of random polynomials. Preprint, 2017. Available at arXiv:1709.08961.
  • [8] I. Benjamini, G. Kalai and O. Schramm. Noise sensitivity of Boolean functions and applications to percolation. Publ. Math. 90 (1) (1999) 5–43.
  • [9] I. Benjamini and O. Schramm. Conformal invariance of Voronoi percolation. Comm. Math. Phys. 197 (1) (1998) 75–107.
  • [10] A. Berlinet and C. Thomas-Agnan. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Springer, 2004.
  • [11] E. D. Bernardino, A. Estrade and J. León. A test of gaussianity based on the Euler characteristic of excursion sets. Electron. J. Stat. 11 (1) (2017) 843–890.
  • [12] E. Bogomolny, R. Dubertrand and C. Schmit. SLE description of the nodal lines of random wavefunctions. J. Phys. A: Math. Theor. 40 (2007) 381–395.
  • [13] E. Bogomolny and C. Schmit. Random wavefunctions and percolation. J. Phys. A: Math. Theor. 40 (2007) 14033–14043.
  • [14] B. Bollobás, O. Riordan. Percolation. Cambridge University Press, Cambridge, 2006.
  • [15] F. Camia and C. M. Newman. Critical percolation exploration path and $\mathit{SLE}_{6}$: A proof of convergence. Probab. Theory Related Fields 139 (3–4) (2007) 473–519.
  • [16] J. Cuzick. A central limit theorem for the number of zeros of a stationary Gaussian process. Ann. Probab. 4 (1976) 547–556.
  • [17] H. Duminil-Copin, A. Raoufi and V. Tassion Subcritical phase of $d$-dimensional Poisson–Boolean percolation and its vacant set. Preprint, 2018. Available at arXiv:1805.00695.
  • [18] H. Duminil-Copin, A. Raoufi and V. Tassion. Exponential decay of connection probabilities for subcritical Voronoi percolation in $\mathbb{R}^{d}$. Probab. Theory Related Fields 173 (1–2) (2019) 479–490.
  • [19] H. Duminil-Copin, A. Raoufi and V. Tassion. Sharp phase transition for the random-cluster and Potts models via decision trees. Ann. of Math. 189 (1) (2019) 75–99.
  • [20] C. Garban and J. Steif. Noise Sensitivity of Boolean Functions and Percolation. Cambridge University Press, Cambridge, 2014.
  • [21] B. Graham and G. Grimmett. Influence and sharp-threshold theorems for monotonic measures. Ann. Probab. 34 (2006) 1726–1745.
  • [22] G. Grimmett. Percolation. Springer, Berlin, Germany, 1999.
  • [23] T. Harris. A lower bound for the critical probability in a certain percolation process. Proc. Camb. Philos. Soc. 56 (1960) 13–20.
  • [24] D. Higdon. Space and space-time modeling using process convolutions. In Quantitative Methods for Current Environmental Issues, C. Anderson, V. Barnett, P. Chatwin and A. El-Shaarawi (Eds). Spring, London, 2002.
  • [25] S. Janson. Gaussian Hilbert Spaces. Cambridge University Press, Cambridge, 1997.
  • [26] H. Kesten. The critical probability of bond percolation on the square lattice equals $1/2$. Comm. Math. Phys. 74 (1980) 41–59.
  • [27] H. Kesten. Scaling relations for 2d-percolation. Comm. Math. Phys. 109 (1) (1987) 109–156.
  • [28] G. Kimeldorf and G. Wahba. A correspondence between Bayesian estimation on stochastic processes and smoothing by splines. Ann. Math. Stat. 41 (2) (1970) 495–502.
  • [29] T. Malevich. Asymptotic normality of the number of crossing of level zero by a Gaussian process. Theory Probab. Appl. 14 (2) (1969) 287–295.
  • [30] S. Molchanov and A. Stepanov. Percolation in random fields. I. Theoret. Math. Phys. 55 (2) (1983) 478–484.
  • [31] S. Molchanov and A. Stepanov. Percolation in random fields. II. Theoret. Math. Phys. 55 (3) (1983) 592–599.
  • [32] S. Molchanov and A. Stepanov. Percolation in random fields. III. Theoret. Math. Phys. 67 (2) (1986) 434–439.
  • [33] F. Nazarov and M. Sodin. Fluctuations in random complex zeroes: Asymptotic normality revisited. Int. Math. Res. Not. 2011 (24) (2011) 5720–5759.
  • [34] F. Nazarov and M. Sodin. Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions. Math. Phys. Anal. Geom. 12 (3) (2016) 205–278.
  • [35] F. Nazarov, M. Sodin and A. Volberg. Transportation to random zeroes by the gradient flow. Geom. Funct. Anal. 17 (3) (2007) 887–935.
  • [36] F. Nazarov, M. Sodin and A. Volberg. The Jancovici–Lebowitz–Manificat law for large fluctuations of random complex zeroes. Comm. Math. Phys. 284 (3) (2008) 833–865.
  • [37] R. O’Donnell, M. Saks, O. Schramm and R. Servedio. Every decision tree has an influential variable. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) 31–39, 2005.
  • [38] L. Pitt. Positively correlated normal variables are associated. Ann. Probab. 10 (2) (1982) 496–499.
  • [39] A. Poularikas. The Handbook of Formulas and Tables for Signal Processing. CRC Press, Boca Raton, 1999.
  • [40] C. Rasmussen and C. Williams. Gaussian Processes for Machine Learning. MIT Press, Cambridge, 2006.
  • [41] A. Rivera and H. Vanneuville. The critical threshold for Bargmann–Fock percolation. Ann. H. Lebesgue To appear, 2019. Available at arXiv:1711.05012.
  • [42] A. Rivera and H. Vanneuville. Quasi-independence for nodal lines. Ann. Inst. H. Poincaré Probab. Statist. To appear, 2019. Available at arXiv:1711.05009.
  • [43] P. Rodriguez. A 0-1 law for the massive Gaussian free field. Probab. Theory Related Fields 169 (2017) 901–930.
  • [44] O. Schramm and S. Sheffield. Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 (1) (2009) 21–137.
  • [45] O. Schramm and J. Steif. Quantitative noise sensitivity and exceptional times for percolation. Ann. of Math. 171 (2) (2010) 619–672.
  • [46] S. Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. 147 (1) (2009) 79–129.
  • [47] S. Smirnov. Towards conformal invariance of $2d$ lattice models. In Proceedings of the ICM, 2007.
  • [48] S. Smirnov and W. Werner. Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 (5–6) (2001) 729–744.
  • [49] V. Tassion. Crossing probabilities for Voronoi percolation. Ann. Probab. 44 (5) (2016) 3385–3398.
  • [50] A. Weinrib. Percolation threshold of a two-dimensional continuum system. Phys. Rev. B 26 (3) (1982) 1352–1361.
  • [51] A. Weinrib. Long-range correlated percolation. Phys. Rev. B 29 (1) (1984) 387–395.
  • [52] H. Wendland. Scattered Data Approximation. Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge, 2005.