TY - JOUR
T1 - Tightness of the recentered maximum of the two-dimensional discrete Gaussian free field
AU - Bramson, Maury
AU - Zeitouni, Ofer
PY - 2012/1
Y1 - 2012/1
N2 - 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.
AB - 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.
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U2 - 10.1002/cpa.20390
DO - 10.1002/cpa.20390
M3 - Article
AN - SCOPUS:80054832898
SN - 0010-3640
VL - 65
SP - 1
EP - 20
JO - Communications on Pure and Applied Mathematics
JF - Communications on Pure and Applied Mathematics
IS - 1
ER -