Communications in Mathematical Physics

, Volume 362, Issue 2, pp 513–546 | Cite as

The Sign Clusters of the Massless Gaussian Free Field Percolate on \({\mathbb{Z}^{d}, d \geqslant 3}\) (and more)

  • Alexander Drewitz
  • Alexis Prévost
  • Pierre-Françcois Rodriguez


We investigate the percolation phase transition for level sets of the Gaussian free field on \({\mathbb{Z}^{d}}\), with \({d \geqslant 3}\), and prove that the corresponding critical parameter h*(d) is strictly positive for all \({d \geqslant 3}\), thus settling an open question from (Rodriguez and Sznitman in Commun Math Phys 320(2):571–601, 2013). In particular, this implies that the sign clusters of the Gaussian free field percolate on \({\mathbb{Z}^{d}}\), for all \({d \geqslant 3}\). Among other things, our construction of an infinite cluster above small, but positive level h involves random interlacements at level u > 0, a random subset of \({\mathbb{Z}^{d}}\) with desirable percolative properties, introduced in Sznitman (Ann Math (2) 171(3):2039–2087, 2010) in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field (Sznitman in Electron Commun Probab 17(9):9, 2012), and a recent coupling of these two objects (Lupu in Ann Probab 44(3):2117–2146, 2016), lifted to a continuous metric graph structure over \({\mathbb{Z}^{d}}\).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abächerli A., Sznitman A.-S.: Level-set percolation for the Gaussian free field on a transient tree. Ann. Inst. Henri Poincarérobab. Stat. 54(1), 173–201 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borodin A.N., Salminen P.: Handbook of Brownian Motion—Facts and Formulae. Probability and Its Applications. Birkhäuser, Basel (2002)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bricmont J., Lebowitz J.L., Maes C.: Percolation in strongly correlated systems: the massless Gaussian field. J. Stat. Phys. 48(5-6), 1249–1268 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brydges D., Fröhlich J., Spencer T.: The random walk representation of classical spin systems and correlation inequalities. Commun. Math. Phys. 83(1), 123–150 (1982)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Campanino M., Russo L.: An upper bound on the critical percolation probability for the three dimensional cubic lattice. Ann. Probab. 13(2), 478–491 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Drewitz A., Ráth B., Sapozhnikov A.: An Introduction to Random Interlacements Springer Briefs in Mathematics. Springer, Berlin (2014)zbMATHGoogle Scholar
  7. 7.
    Drewitz, A., Ráth, B., Sapozhnikov, A.: On chemical distances and shape theorems in percolation models with long-range correlations. J. Math. Phys. 55(8), 083307, 30 (2014)Google Scholar
  8. 8.
    Drewitz A., Rodriguez P.-F.: High-dimensional asymptotics for percolation of Gaussian free field level sets. Electron. J. Probab. 20(47), 39 (2015)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dynkin E.B.: Markov processes as a tool in field theory. J. Func. Anal. 50(2), 167–187 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eisenbaum N., Kaspi H., Marcus M.B., Rosen J., Shi Z.: A Ray–Knight theoremfor symmetric Markov processes. Ann. Probab. 28(4), 1781–1796 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Enriquez N., Kifer Y.: Markov chains on graphs and Brownian motion. J. Theor. Probab. 14(2), 495–510 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Folz M.: Volume growth and stochastic completeness of graphs. Trans. Am. Math. Soc. 366(4), 2089–2119 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lupu T.: From loop clusters and random interlacements to the free field. Ann. Probab. 44(3), 2117–2146 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lupu, T., Sabot, C., Tarrès, P.: Inverting the coupling of the signed Gausssian free field with a loop soup. Preprint, arXiv:1701.01092 (2017)
  15. 15.
    Marcus M.B., Rosen J.: Markov Processes, Gaussian Processes, and Local Times, Volume 100 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  16. 16.
    Marinov, V.: Percolation in Correlated Systems. Ph.D. thesis. Rutgers University (2007)Google Scholar
  17. 17.
    Popov S., Ráth B.: On decoupling inequalities and percolation of excursion sets of the Gaussian free field. J. Stat. Phys. 159(2), 312–320 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Popov S., Teixeira A.: Soft local times and decoupling of random interlacements. J. Eur. Math. Soc. (JEMS) 17(10), 2545–2593 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ráth B., Sapozhnikov A.: On the transience of random interlacements. Electron. Commun. Probab. 16, 379–391 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ráth B., Sapozhnikov A.: The effect of small quenched noise on connectivity properties of random interlacements. Electron. J. Probab. 18(4), 20 (2013)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Revuz D., Yor M.: Continuous Martingales and Brownian Motion, Volume 293 of Grundlehren der Mathematischen Wissenschaften, 3rd edition. Springer, Berlin (1999)CrossRefGoogle Scholar
  22. 22.
    Rodriguez P.-F.: Level set percolation for random interlacements and the Gaussian free field. Stoch. Process. Appl. 124(4), 1469–1502 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rodriguez, P.-F.: Decoupling inequalities for the Ginzburg-Landau \({\nabla_\varphi}\) models. Preprint, arXiv:1612.02385 (2016)
  24. 24.
    Rodriguez P.-F., Sznitman A.-S.: Phase transition and level-set percolation for the Gaussian free field. Commun. Math. Phys. 320(2), 571–601 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Symanzik, K.: Euclidean quantum field theory. In: Scuola internazionale di Fisica “Enrico Fermi”. XLV Corso. Academic Press (1969)Google Scholar
  26. 26.
    Sznitman A.-S.: Vacant set of random interlacements and percolation. Ann. Math. (2) 171(3), 2039–2087 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sznitman A.-S.: Decoupling inequalities and interlacement percolation on \({G \times \mathbb{Z}}\). Invent. Math. 187(3), 645–706 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sznitman A.-S.: An isomorphism theorem for random interlacements. Electron. Commun. Probab. 17(9), 9 (2012)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sznitman A.-S.: Random interlacements and the Gaussian free field. Ann. Probab. 40(6), 2400–2438 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sznitman A.-S.: Topics in Occupation Times and Gaussian Free Fields Zurich Lectures in Advanced Mathematics.. European Mathematical Society (EMS), Z00FC;rich (2012)CrossRefzbMATHGoogle Scholar
  31. 31.
    Sznitman A.-S.: Disconnection and level-set percolation for the Gaussian free field. J. Math. Soc. Jpn. 67(4), 1801–1843 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Sznitman A.-S.: Coupling and an application to level-set percolation of the Gaussian free field. Electron. J. Probab. 21(35), 26 (2016)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnCologneGermany
  2. 2.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA

Personalised recommendations