The Annals of Probability

Extremes of the discrete two-dimensional Gaussian free field

Olivier Daviaud

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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.

Article information

Ann. Probab., Volume 34, Number 3 (2006), 962-986.

First available in Project Euclid: 27 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60G15: Gaussian processes 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Free field extrema of Gaussian fields entropic repulsion multiscale decomposition large deviations


Daviaud, Olivier. Extremes of the discrete two-dimensional Gaussian free field. Ann. Probab. 34 (2006), no. 3, 962--986. doi:10.1214/009117906000000061.

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