Sharp phase transition for Gaussian percolation in all dimensions
Annales Henri Lebesgue, Volume 5 (2022), pp. 987-1008.


Keywords percolation, sharpness, phase transition, Gaussian fields


We consider the level-sets of continuous Gaussian fields on d above a certain level -, which defines a percolation model as varies. We assume that the covariance kernel satisfies certain regularity, symmetry and positivity conditions as well as a polynomial decay with exponent greater than d (in particular, this includes the Bargmann–Fock field). Under these assumptions, we prove that the model undergoes a sharp phase transition around its critical point c . More precisely, we show that connection probabilities decay exponentially for < c and percolation occurs in sufficiently thick 2D slabs for > c . This extends results recently obtained in dimension d=2 to arbitrary dimensions through completely different techniques. The result follows from a global comparison with a truncated (i.e. with finite range of dependence) and discretized (i.e. defined on the lattice ε d ) version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a slight change in the parameter .


[AB87] Aizenman, Michael; Barsky, David J. Sharpness of the phase transition in percolation models, Commun. Math. Phys., Volume 108 (1987) no. 3, pp. 489-526 | DOI | MR | Zbl

[Ana15] Anantharaman, Nalini Topologie des hypersurfaces nodales de fonctions gaussiennes, Séminaire Bourbaki Vol. 2015/2016 (Astérisque), Volume 390, Société Mathématique de France, 2015, pp. 369-408 | Zbl

[Bar04] Barlow, Martin T. Random walks on supercritical percolation clusters, Ann. Probab., Volume 32 (2004) no. 4, pp. 3024-3084 | MR | Zbl

[BG17] Beffara, Vincent; Gayet, Damien Percolation of random nodal lines, Publ. Math., Inst. Hautes Étud. Sci., Volume 126 (2017) no. 1, pp. 131-176 | DOI | MR | Zbl

[BM18] Beliaev, Dmitri B.; Muirhead, Stephen Discretisation schemes for level sets of planar gaussian fields, Commun. Math. Phys., Volume 359 (2018) no. 3, pp. 869-913 | DOI | MR | Zbl

[BS02] Bogomolny, Eugene; Schmit, Charles Percolation model for nodal domains of chaotic wave functions, Phys. Rev. Lett., Volume 88 (2002) no. 11, 114102, 4 pages | DOI

[BS07] Bogomolny, Eugene; Schmit, Charles Random wavefunctions and percolation, J. Phys. A, Math. Theor., Volume 40 (2007) no. 47, pp. 14033-14043 | DOI | MR | Zbl

[BT17] Benjamini, Itai; Tassion, Vincent Homogenization via sprinklin, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 53 (2017) no. 2, pp. 997-1005 | DOI | Zbl

[Cer00] Cerf, Raphaël Large deviations for three dimensional supercritical percolation, Astérisque, 267, Société Mathématique de France, 2000 | Numdam | Zbl

[CS19] Canzani, Yaiza; Sarnak, Peter Topology and nesting of the zero set components of monochromatic random waves, Commun. Pure Appl. Math., Volume 72 (2019) no. 2, pp. 343-374 | DOI | MR | Zbl

[DCGRS20] Duminil-Copin, Hugo; Goswami, Subhajit; Rodriguez, Pierre-François; Severo, Franco Equality of critical parameters for percolation of Gaussian free field level-sets (2020) (

[DCRT19] Duminil-Copin, Hugo; Raoufi, Aran; Tassion, Vincent Sharp phase transition for the random-cluster and potts models via decision trees, Ann. Math., Volume 189 (2019) no. 1, pp. 75-99 | MR | Zbl

[DM21] Dewan, Vivek; Muirhead, Stephen Upper bounds on the one-arm exponent for dependent percolation models (2021) (

[DPR21] Drewitz, Alexander; Prévost, Alexis; Rodriguez, Pierre-François Critical exponents for a percolation model on transient graph (2021) (

[GM90] Grimmett, Geoffrey R.; Marstrand, John M. The supercritical phase of percolation is well behaved, Proc. R. Soc. Lond., Ser. A, Volume 430 (1990) no. 1879, pp. 439-457 | MR | Zbl

[Gri99] Grimmett, Geoffrey R. Percolation, Grundlehren der Mathematischen Wissenschaften, 321, Springer, 1999 | DOI | Zbl

[GRS21] Goswami, Subhajit; Rodriguez, Pierre-François; Severo, Franco On the radius of Gaussian free field excursion clusters (2021) (

[Jan97] Janson, Svante Gaussian Hilbert Spaces, Cambridge Tracts in Mathematics, 129, Cambridge University Press, 1997 | DOI | Zbl

[Kes80] Kesten, Harry The critical probability of bond percolation on the square lattice equals 1 2, Commun. Math. Phys., Volume 74 (1980) no. 1, pp. 41-59 | DOI | MR | Zbl

[Men86] Menshikov, Mikhail V. Coincidence of critical points in percolation problems, Dokl. Akad. Nauk SSSR, Volume 288 (1986) no. 6, pp. 1308-1311 | MR | Zbl

[MRVKS20] Muirhead, Stephen; Rivera, Alejandro; Vanneuville, Hugo; Köhler-Schindler, Laurin The phase transition for planar gaussian percolation models without FKG (2020) (

[MS83a] Molchanov, Stanislav A.; Stepanov, A. K. Percolation in random fields. I, Teor. Mat. Fiz., Volume 55 (1983) no. 2, pp. 246-256 | MR

[MS83b] Molchanov, Stanislav A.; Stepanov, A. K. Percolation in random fields. II, Teor. Mat. Fiz., Volume 55 (1983) no. 3, pp. 592-599 | DOI | MR

[MS86] Molchanov, Stanislav A.; Stepanov, A. K. Percolation in random fields. III, Teor. Mat. Fiz., Volume 67 (1986) no. 2, pp. 434-439 | DOI | MR

[MV20] Muirhead, Stephen; Vanneuville, Hugo The sharp phase transition for level set percolation of smooth planar Gaussian fields, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 56 (2020) no. 2, pp. 1358-1390 | MR | Zbl

[Nit18] Nitzschner, Maximilian Disconnection by level sets of the discrete Gaussian free field and entropic repulsion, Electron. J. Probab., Volume 23 (2018) no. 1, 105 | MR | Zbl

[NS09] Nazarov, Fedor; Sodin, Mikhail On the number of nodal domains of random spherical harmonics, Am. J. Math., Volume 131 (2009) no. 5, pp. 1337-1357 | DOI | MR | Zbl

[NS20] Nitzschner, Maximilian; Sznitman, Alain-Sol Solidification of porous interfaces and disconnection, J. Eur. Math. Soc., Volume 22 (2020) no. 8, pp. 2629-2672 | DOI | MR | Zbl

[PR15] Popov, Serguei; Ráth, Balázs On decoupling inequalities and percolation of excursion sets of the Gaussian free field, J. Stat. Phys., Volume 159 (2015) no. 2, pp. 312-320 | DOI | MR | Zbl

[Riv19] Rivera, Alejandro Talagrand’s inequality in planar gaussian field percolation (2019) (

[RV20] Rivera, Alejandro; Vanneuville, Hugo The critical threshold for Bargmann–Fock percolation, Ann. Henri Lebesgue, Volume 3 (2020), pp. 169-215 | DOI | MR | Zbl

[Sap17] Sapozhnikov, Artem Random walks on infinite percolation clusters in models with long-range correlations, Ann. Probab., Volume 45 (2017) no. 3, pp. 1842-1898 | MR | Zbl

[Sar17] Sarnak, Peter Topologies of the zero sets of random real projective hyper-surfaces and of monochromatic waves, 2017 (Talk delivered at Random geometries/ Random topologies conference, slides available at

[SW19] Sarnak, Peter; Wigman, Igor Topologies of nodal sets of random band-limited functions, Commun. Pure Appl. Math., Volume 72 (2019) no. 2, pp. 275-342 | DOI | MR | Zbl

[Szn15] Sznitman, Alain-Sol Disconnection and level-set percolation for the Gaussian free field, J. Math. Soc. Japan, Volume 67 (2015) no. 4, pp. 1801-1843 | MR | Zbl