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Hölder regularity and dimension bounds for random curves. (English) Zbl 0944.60022
Consider for every \(\delta\in ]0,1]\) a probability measure \(\gamma_\delta\) on a closed set of polygonal paths of step-size \(\delta\), contained in some fixed compact subset of \(\mathbb{R}^d\). The aim of this work is to yield general criteria giving information on the existence and the support of weak limits of \(\gamma_\delta\), as \(\delta\downarrow 0\). The first criterion requires a uniform control of multiple crossings of spherical shells by the paths, and guarantees the existence of some weak limit of \(\gamma_\delta\), supported by curves having a common upper bound on their Hausdorff dimension. The second criterion requires a uniform control on simultaneous crossings of separated cylinders by the paths, and guarantees a uniform lower bound, strictly larger than 1, for the Hausdorff dimension of the curves supporting any weak limit of \(\gamma_\delta\). Main examples are given by percolation models and random spanning trees.
Reviewer: J.Franchi (Paris)

MSC:
60D05 Geometric probability and stochastic geometry
28A75 Length, area, volume, other geometric measure theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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