Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field
Article first published online: 15 SEP 2011
Copyright © 2011 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 65, Issue 1, pages 1–20, January 2012
How to Cite
Bramson, M. and Zeitouni, O. (2012), Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field. Comm. Pure Appl. Math., 65: 1–20. doi: 10.1002/cpa.20390
- Issue published online: 18 OCT 2011
- Article first published online: 15 SEP 2011
- Manuscript Received: SEP 2010
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. © 2011 Wiley Periodicals, Inc.