Tightness of the recentered maximum of the two‐dimensional discrete Gaussian free field
By Maury Bramson, Ofer Zeitouni. 2010
126 citations recorded by the source corpus.
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.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/9374f8fa960cc5422f6fce8a8c0fa0a94face47a
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In Statistics and Probability
Citations in the corpus
- Convergence of the centered maximum of log-correlated Gaussian fields (2015, methodology): “Those properties are: (a) The tightness of MN −mN , and the bounds on the right and left tails of MN −mN as well as certain geometric properties of maxima for the log-correlated fields under consideration, follow from modifying arguments in [9, 11, 10].”
- The subleading order of two dimensional cover times (2014, methodology): “Thus the subleading correction for BBM (and by extension the Gaussian free field on the two dimensional torus [7]) correctly “predicts” the correction term of the ε-cover time of the torus, once this small difference in the tail is taken into account.”
- Maximum of a log-correlated Gaussian field (2013, background): “Let us mention that in the discrete setting, Z2 ∩ [0,N ]2, if we add the zero boundary condition, the model becomes the so-called Gaussian free field (GFF), which has attracted many recent attentions, see [15], [14] and [17].”
- Asymptotics of cover times via Gaussian free fields: Bounded-degree graphs and general trees (2011, methodology): “For the 2D lattice, in a recent breakthrough paper of Bramson and Zeitouni [11], it was shown that the supremum of the Gaussian free field is tight after proper centering, together with an estimate on its expectation up to an additive constant.”
- Critical Gaussian multiplicative chaos: Convergence of the derivative martingale (2012, background): “Let us mention the glassy phase of log-correlated random potentials (see [6, 19, 36, 37]) or the asymptotic expansion of the maximum of log-correlated random variables; see [17, 24].”
- Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field (2013, methodology): “N mN / (see [11]), it suffices to show that, for any fixed x 2 R, (5.”
- Extreme value statistics of correlated random variables: A pedagogical review (2019, background): “Since then, these models have generated a lot of interest, both in physics and in mathematics, because of their relations with the extremes of the two-dimensional Gaussian Free Field (2d-GFF) [74, 243, 244].”
- The extremal process of branching Brownian motion (2011, background): “ passing that a similar structure is expected to emerge in all the models which are conjectured to fall into the universality class of branching Brownian motion, such as the 2-dim Gaussian Free Field [7, 8, 14], or the cover time for the simple random walk on the two dimensional discrete torus [20, ”
- Extreme values for two-dimensional discrete Gaussian free field (2012, methodology): “It is worthwhile emphasizing that the precise estimate on the expectation of the maximum of the GFF in [17] plays a crucial role in [24].”