Tightness of the recentered maximum of log-correlated Gaussian fields

Javier Acosta (University of Minnesota)

Abstract


We consider a family of centered Gaussian fields on the  d-dimensional unit box, whose covariance decreases logarithmically in the distance between points. We prove tightness of the recentered maximum of the Gaussian fields and provide exponentially decaying bounds on the right and left tails. We then apply this result to a version of the two-dimensional continuous Gaussian free field.

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

Pages: 1-25

Publication Date: October 3, 2014

DOI: 10.1214/EJP.v19-3170

References

  • Krylov, N. V. Introduction to the theory of random processes. Graduate Studies in Mathematics, 43. American Mathematical Society, Providence, RI, 2002. xii+230 pp. ISBN: 0-8218-2985-8 MR1885884
  • Adler, Robert J.; Taylor, Jonathan E. Random fields and geometry. Springer Monographs in Mathematics. Springer, New York, 2007. xviii+448 pp. ISBN: 978-0-387-48112-8 MR2319516
  • Bramson, Maury; Zeitouni, Ofer. Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math. 65 (2012), no. 1, 1--20. MR2846636
  • Ding, Jian. Exponential and double exponential tails for maximum of two-dimensional discrete Gaussian free field. Probab. Theory Related Fields 157 (2013), no. 1-2, 285--299. MR3101848 http://arxiv.org/abs/1105.5833
  • M. Bramson, J. Ding and O. Zeitouni. Convergence in law of the maximum of the two-dimensional discrete Gaussian free field, 2013. http://arxiv.org/abs/1301.6669
  • 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
  • Daviaud, Olivier. Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34 (2006), no. 3, 962--986. MR2243875
  • Hu, Xiaoyu; Miller, Jason; Peres, Yuval. Thick points of the Gaussian free field. Ann. Probab. 38 (2010), no. 2, 896--926. MR2642894
  • Dynkin, E. B. Markov processes and random fields. Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 3, 975--999. MR0585179
  • Sheffield, Scott. Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 (2007), no. 3-4, 521--541. MR2322706
  • Duplantier, Bertrand; Rhodes, Remi; Sheffield, Scott; Vargas, Vincent. Critical Gaussian multiplicative chaos: Convergence of the derivative martingale. Ann. Probab. 42 (2014), no. 5, 1769--1808. MR3262492 http://arxiv.org/abs/1206.1671
  • T. Madaule. Maximum of a log-correlated Gaussian field, 2013. http://arxiv.org/abs/1307.1365
  • T. Madaule, R. Rhodes and V. Vargas. Glassy phase and freezing of log-correlated Gaussian potentials. http://arxiv.org/abs/1310.5574
  • Adler, Robert J. An introduction to continuity, extrema, and related topics for general Gaussian processes. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 12. Institute of Mathematical Statistics, Hayward, CA, 1990. x+160 pp. ISBN: 0-940600-17-X MR1088478
  • Ding, Jian; Zeitouni, Ofer. Extreme values for two-dimensional discrete Gaussian free field. Ann. Probab. 42 (2014), no. 4, 1480--1515. MR3262484 http://arxiv.org/abs/1206.0346
  • Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1988. xxiv+470 pp. ISBN: 0-387-96535-1 MR0917065


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