Tightness of the recentered maximum of the two‐dimensional discrete Gaussian free field
Top Cited Papers
- 15 September 2011
- journal article
- research article
- Published by Wiley in Communications on Pure and Applied Mathematics
- Vol. 65 (1) , 1-20
Abstract
We consider the maximum of the discrete two‐dimensional Gaussian free field (GFF) in a box and prove that its maximum, centered at its mean, is tight, settling a longstanding conjecture. The proof combines a recent observation by Bolthausen, Deuschel, and Zeitouni with elements from Bramson's results on branching Brownian motion and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two‐dimensional torus, are also discussed.Keywords
All Related Versions
This publication has 10 references indexed in Scilit:
- Recursions and tightness for the maximum of the discrete, two dimensional Gaussian Free FieldElectronic Communications in Probability, 2011
- Thick points of the Gaussian free fieldThe Annals of Probability, 2010
- Minima in branching random walksThe Annals of Probability, 2009
- Extremes of the discrete two-dimensional Gaussian free fieldThe Annals of Probability, 2006
- Entropic Repulsion and the Maximum of the two-dimensional harmonicThe Annals of Probability, 2001
- Glass transition of a particle in a random potential, front selection in nonlinear renormalization group, and entropic phenomena in Liouville and sinh-Gordon modelsPhysical Review E, 2001
- Limit distributions for minimal displacement of branching random walksProbability Theory and Related Fields, 1991
- Intersections of Random Walks1991
- An introduction to continuity, extrema, and related topics for general Gaussian processes1990
- Maximal displacement of branching brownian motionCommunications on Pure and Applied Mathematics, 1978