Random soups, carpets and fractal dimensions. (English) Zbl 1223.28012

The authors study some elementary properties of a class of random connected planar fractal sets obtained by removing from \([0,1]^2\) all squares of a statistically translation-invariant and scale-invariant Poisson point process of squares. The Brownian loop-soup, which was introduced by G. F. Lawler and W. Werner [Probab. Theory Relat. Fields 128, No. 4, 565–588 (2004; Zbl 1049.60072)], is the random carpet obtained by removing the interiors of a Poisson collection of Brownian loops.
The main result of the paper is to show that the Hausdorff dimensions of random carpets defined by a random soup are deterministic and equal to their expectation dimension. The techniques are based on the second-moment method. This result applies to the determination of the Hausdorff dimension of the conformal loop ensembles (CLE) defined via Poissonian clouds of Brownian loops, for which the expectation dimension is computed by O. Schramm, S. Sheffield and D. B. Wilson [Commun. Math. Phys. 288, No. 1, 43–53 (2009; Zbl 1187.82044)].


28A80 Fractals
82B43 Percolation
28A78 Hausdorff and packing measures
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