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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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