Recursions and tightness for the maximum of the discrete, two dimensional Gaussian Free Field

Erwin Bolthausen (Universität Zürich)
Jean-Dominique Deuschel (Technische Universität Berlin)
Ofer Zeitouni (University of Minnesota)

Abstract


We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is tight. The method of proof relies on an argument developed by Dekking and Host for branching random walks with bounded increments and on comparison results specific to Gaussian fields.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 114-119

Publication Date: February 16, 2011

DOI: 10.1214/ECP.v16-1610

References

  1. Addario-Berry, Louigi; Reed, Bruce. Minima in branching random walks. Ann. Probab. 37 (2009), no. 3, 1044--1079. MR2537549 (2011b:60338)

  2. Bolthausen, Erwin; Deuschel, Jean-Dominique; Giacomin, Giambattista. Entropic repulsion and the maximum of the two-dimensional harmonic crystal. Ann. Probab. 29 (2001), no. 4, 1670--1692. MR1880237 (2003a:82028)

  3. Bolthausen, Erwin; Deuschel, Jean-Dominique; Zeitouni, Ofer. Recursions and tightness for the maximum of the discrete, two dimensional gaussian free field. (2010). arXiv:1005.5417v1

  4. Bramson, Maury; Zeitouni, Ofer. Tightness for the minimal displacement of branching random walk. J. Stat. Mech. Theory Exp. 2007, no. 7, P07010, 12 pp. (electronic). MR2335694 (2008g:60255)

  5. Bramson, Maury; Zeitouni, Ofer. Tightness for a family of recursion equations. Ann. Probab. 37 (2009), no. 2, 615--653. MR2510018 (2010c:60248)

  6. Bramson, Maury; Zeitouni, Ofer. Tightness of the recentered maximum of the two-dimensional discrete Gaussian Free Field. (2010). arXiv:1009.3443v1

  7. Carpentier, D.; Le Doussal, P.. Glass transition for a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in liouville and sinh-gordon models. Phys. Rev. E. 63 (2001), 026110.

  8. Chatterjee, Sourav. A new method of normal approximation. Ann. Probab. 36 (2008), no. 4, 1584--1610. MR2435859 (2009j:60043)

  9. Daviaud, Olivier. Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34 (2006), no. 3, 962--986. MR2243875 (2007m:60289)

  10. Dekking, F. M.; Host, B. Limit distributions for minimal displacement of branching random walks. Probab. Theory Related Fields 90 (1991), no. 3, 403--426. MR1133373 (93b:60189)

  11. Le Doussal, P.; Fyodorov, Y.; Rosso, A. newblock Statistical mechanics of logarithmic rem: duality, freezing and extreme value statistics of $1/f$ noises generated by gaussian free fields. J. Stat. Mech. (2009), P10005.

  12. Fyodorov, Yan V.; Bouchaud, Jean-Philippe. Freezing and extreme-value statistics in a random energy model with logarithmically correlated potential. J. Phys. A 41 (2008), no. 37, 372001, 12 pp. MR2430565 (2010g:82036)

  13. Hu, Xiaoyu; Miller, Jason; Peres, Yuval. Thick points of the Gaussian free field. Ann. Probab. 38 (2010), no. 2, 896--926. MR2642894

  14. Sheffield, Scott. Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 (2007), no. 3-4, 521--541. MR2322706 (2008d:60120)

Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.