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Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field

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Abstract

We consider the discrete Gaussian Free Field in a square box in \({\mathbb{Z}^2}\) of side length N with zero boundary conditions and study the joint law of its properly-centered extreme values (h) and their scaled spatial positions (x) in the limit as \({N \to \infty}\). Restricting attention to extreme local maxima, i.e., the extreme points that are maximal in an r N -neighborhood thereof, we prove that the associated process tends, whenever \({r_N \to \infty}\) and \({r_N/N \to 0}\), to a Poisson point process with intensity measure \({Z{(\rm dx)}{\rm e}^{-\alpha h} {\rm d}h}\), where \({\alpha:= 2/\sqrt{g}}\) with g: = 2/π and where Z(dx) is a random Borel measure on [0, 1]2. In particular, this yields an integral representation of the law of the absolute maximum, similar to that found in the context of Branching Brownian Motion. We give evidence that the random measure Z is a version of the derivative martingale associated with the continuum Gaussian Free Field.

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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, USA

    Marek Biskup & Oren Louidor

  2. Faculty of Industrial Engineering and Management, Technion, Haifa, Israel

    Oren Louidor

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  1. Marek Biskup
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  2. Oren Louidor
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Correspondence to Marek Biskup.

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Communicated by F. Toninelli

© 2016 by M. Biskup and O. Louidor. Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.

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Biskup, M., Louidor, O. Extreme Local Extrema of Two-Dimensional Discrete Gaussian Free Field. Commun. Math. Phys. 345, 271–304 (2016). https://doi.org/10.1007/s00220-015-2565-8

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