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A 0–1 law for the massive Gaussian free field

Abstract

We investigate the phase transition in a non-planar correlated percolation model with long-range dependence, obtained by considering level sets of a Gaussian free field with mass above a given height h. The dependence present in the model is a notorious impediment when trying to analyze the behavior near criticality. Alongside the critical threshold \(h_*\) for percolation, a second parameter \(h_{**} \ge h_*\) characterizes a strongly subcritical regime. We prove that the relevant crossing probabilities converge to 1 polynomially fast below \(h_{**}\), which (firmly) suggests that the phase transition is sharp. A key tool is the derivation of a suitable differential inequality for the free field that enables the use of a (conditional) influence theorem.

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References

  1. Aizenman, M., Barsky, D.J.: Sharpness of the phase transition in percolation models. Commun. Math. Phys. 108(3), 489–526 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Beffara, V., Duminil-Copin, H.: The self-dual point of the two-dimensional random-cluster model is critical for \(q\ge 1\). Probab. Theory Relat. Fields 153(3–4), 511–542 (2012)

    Article  MATH  Google Scholar 

  3. Bollobás, B., Riordan, O.: The critical probability for random Voronoi percolation in the plane is 1/2. Probab. Theory Relat. Fields 136(3), 417–468 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bolthausen, E., Deuschel, J.-D., Zeitouni, O.: Entropic repulsion of the lattice free field. Commn. Math. Phys. 170(2), 417–443 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bourgain, J., Kahn, J., Kalai, G., Katznelson, Y., Linial, N.: The influence of variables in product spaces. Isr. J. Math. 77(1–2), 55–64 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Drewitz, A., Rodriguez, P.-F.: High-dimensional asymptotics for percolation of gaussian free field level sets. Electron. J. Probab. 20(47), 1–39 (2015)

    MATH  MathSciNet  Google Scholar 

  8. Duminil-Copin, H., Manolescu, I.: The phase transitions of the planar random-cluster and Potts models with q larger than 1 are sharp. Probab. Theory Relat. Fields 164(3–4), 865–892 (2016)

    Article  MATH  Google Scholar 

  9. Duminil-Copin, H., Tassion, V.: A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. Commun. Math. Phys. 343(2), 725–745 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  10. Garet, O.: Percolation transition for some excursion sets. Electron. J. Probab. 9, 255–292 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Giacomin, G.: Aspects of statistical mechanics of random surfaces. Notes of lectures given at I.H.P. (2001). http://www.proba.jussieu.fr/pageperso/giacomin/pub/IHP.ps

  12. Graham, B.T., Grimmett, G.R.: Influence and sharp-threshold theorems for monotonic measures. Ann. Probab. 34(5), 1726–1745 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grimmett, G.: Percolation, vol. 321 of Grundlehren der Mathematischen Wissenschaften, 2nd edn. Springer, Berlin (1999)

  14. Grimmett, G.: The random-cluster model. Grundlehren der Mathematischen Wissenschaften, vol. 333. Springer, Berlin (2006)

  15. Holley, R.: Remarks on the FKG inequalities. Commun. Math. Phys. 36, 227–231 (1974)

    Article  MathSciNet  Google Scholar 

  16. Kahn, J., Kalai, G., Linial, N.: The influence of variables on boolean functions. In: Foundations of Computer Science, 29th Annual Symposium on, pp. 68–80. IEEE (1988)

  17. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus, vol. 113 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1991)

  18. Lawler, G.F.: Intersections of random walks. Probability and its applications. Birkhäuser Boston Inc, Boston, MA (1991)

    Book  MATH  Google Scholar 

  19. Lebowitz, J.L., Saleur, H.: Percolation in strongly correlated systems. Phys. A 138(1–2), 194–205 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lupu, T.: From loop clusters and random interlacement to the free field (2014). arXiv:1402.0298 (preprint)

  21. Margulis, G.A.: Probabilistic characteristics of graphs with large connectivity. Probl. Peredači Inf. 10(2), 101–108 (1974)

    MATH  MathSciNet  Google Scholar 

  22. Menshikov, M.V.: Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR 288(6), 1308–1311 (1986)

    MathSciNet  Google Scholar 

  23. Molchanov, S.A., Stepanov, A.K.: Percolation in random fields. I. Teoret. Mat. Fiz. 55(2), 246–256 (1983)

    MathSciNet  Google Scholar 

  24. Molchanov, S.A., Stepanov, A.K.: Percolation in random fields. II. Teoret. Mat. Fiz. 55(3), 419–430 (1983)

    MathSciNet  Google Scholar 

  25. 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)

    Article  MATH  MathSciNet  Google Scholar 

  26. Preston, C.J.: A generalization of the FKG inequalities. Commun. Math. Phys. 36, 233–241 (1974)

    Article  MathSciNet  Google Scholar 

  27. Rodriguez, P.-F.: Level set percolation for random interlacements and the Gaussian free field. Stoch. Process. Appl. 124(4), 1469–1502 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  28. 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)

    Article  MATH  MathSciNet  Google Scholar 

  29. Russo, L.: On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56(2), 229–237 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  30. Sznitman, A.-S.: Vacant set of random interlacements and percolation. Ann. Math. 171(3), 2039–2087 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sznitman, A.S.: An isomorphism theorem for random interlacements. Electron. Commun. Probab. 17(9), 1–9 (2012)

  32. Sznitman, A.-S.: Topics in occupation times and Gaussian free fields. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2012)

    Book  MATH  Google Scholar 

  33. Sznitman, A.S.: Disconnection and level-set percolation for the Gaussian free field. J. Math. Soc. Jpn. (Special Issue dedicated to Prof. K. Itô) 67(4), 1801-–1843 (2014)

    MATH  MathSciNet  Google Scholar 

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Acknowledgments

The author thanks Alain-Sol Sznitman for several useful discussions and for his comments on an earlier draft of this manuscript, and both referees for their valuable remarks.

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Correspondence to Pierre-François Rodriguez.

Additional information

This research was supported in part by the Grant ERC-2009-AdG 245728-RWPERCRI.

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Rodriguez, PF. A 0–1 law for the massive Gaussian free field. Probab. Theory Relat. Fields 169, 901–930 (2017). https://doi.org/10.1007/s00440-016-0743-z

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  • DOI: https://doi.org/10.1007/s00440-016-0743-z

Mathematics Subject Classification

  • 60G15
  • 60G60
  • 60K35
  • 82B26
  • 82B27
  • 82B41
  • 82B43