Open Access
May 2006 Extremes of the discrete two-dimensional Gaussian free field
Olivier Daviaud
Ann. Probab. 34(3): 962-986 (May 2006). DOI: 10.1214/009117906000000061


We consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high (or low). We then extend some of our computations to the case of the free field conditioned on being everywhere nonnegative. For example, we compute the width of the largest downward spike of a given length. Through the prism of these results, we find that the extrema of the free field under entropic repulsion (minus its mean) and those of the unconditioned free field are identical. Finally, when compared to previous results these findings reveal a suggestive analogy between the square of the free field and the two-dimensional simple random walk on the discrete torus.


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Olivier Daviaud. "Extremes of the discrete two-dimensional Gaussian free field." Ann. Probab. 34 (3) 962 - 986, May 2006.


Published: May 2006
First available in Project Euclid: 27 June 2006

zbMATH: 1104.60062
MathSciNet: MR2243875
Digital Object Identifier: 10.1214/009117906000000061

Primary: 60G15 , 60K35 , 82B41

Keywords: Entropic repulsion , extrema of Gaussian fields , Free field , large deviations , multiscale decomposition

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 3 • May 2006
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