Extremes of the discrete two-dimensional Gaussian free field

Details Extremes of the discrete two-dimensional Gaussian free field Extremes of the discrete two-dimensional Gaussian free field

2004-06-29

arxiv.org/abs/math/0406609v4

Annals of Probability 2006, Vol. 34, No. 3, 962-986

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117906000000061 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

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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