Open Access
November 2020 Maximum of the Ginzburg–Landau fields
David Belius, Wei Wu
Ann. Probab. 48(6): 2647-2679 (November 2020). DOI: 10.1214/19-AOP1416


We study a two-dimensional massless field in a box with potential $V(\nabla \phi (\cdot ))$ and zero boundary condition, where $V$ is any symmetric and uniformly convex function. Naddaf–Spencer (Comm. Math. Phys. 183 (1997) 55–84) and Miller (Comm. Math. Phys. 308 (2011) 591–639) proved that the rescaled macroscopic averages of this field converge to a continuum Gaussian free field. In this paper, we prove that the distribution of local marginal $\phi (x)$, for any $x$ in the bulk, has a Gaussian tail. We further characterize the leading order of the maximum and the dimension of high points of this field, thus generalizing the results of Bolthausen–Deuschel–Giacomin (Ann. Probab. 29 (2001) 1670–1692) and Daviaud (Ann. Probab. 34 (2006) 962–986) for the discrete Gaussian free field.


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David Belius. Wei Wu. "Maximum of the Ginzburg–Landau fields." Ann. Probab. 48 (6) 2647 - 2679, November 2020.


Received: 1 April 2018; Revised: 1 November 2019; Published: November 2020
First available in Project Euclid: 20 October 2020

MathSciNet: MR4164451
Digital Object Identifier: 10.1214/19-AOP1416

Primary: 60G50 , 60K35

Keywords: extrema of log-correlated field , Ginzburg–Landau field , multiscale decomposition , stochstic interface models

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 6 • November 2020
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