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Extremes of the discrete two-dimensional Gaussian free field. (English) Zbl 1104.60062
A two-dimensional Gaussian free field (GFF) is a family of centered Gaussian random variables $$\Phi =\{\Phi_x; x\in V_N\}$$ with covariance given by $$G_N(x,y)$$, the Green function of the two-dimensional simple random walk in the square $$V_N= [ 1,N]^2\cap \mathbb Z^2$$ stopped at the boundary. The conditioned free field (CFF) is the law of $$\Phi$$ conditioned on being nonnegative.
The first part of the paper contains a precise description of the number of high (resp. low) points of a GFF and the fractal properties of their repartition in $$V_N$$. The proofs use a multiscale decomposition introduced by E. Bolthausen, J.-D. Deuschel and G. Giacomin [Ann. Probab. 29, No. 4, 1670–1692 (2001; Zbl 1034.82018)] who first computed the maximum of a two-dimensional GFF and, in particular, showed that the mean of the CFF is of the order $$\log N$$. The proofs also rely on a close analogy between the GFF and $$\tau(x)$$, the first time $$x$$ is visited by a simple random walk on the torus $$\mathbb Z^2/N\mathbb Z^2$$ starting at 0, which is carefully studied by A. Dembo, Y. Peres, J. Rosen and O. Zeitouni [ibid. 34, No. 1, 219–263 (2006; Zbl 1100.60057)]. These results are then used to study the CFF. One sees that, contrary to the situation in dimensions $$\geq 3$$, the extrema of the shifted GFF and the CFF look alike.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60G15 Gaussian processes 82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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##### References:
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