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Noise sensitivity and Voronoi percolation. (English) Zbl 1402.60122
Summary: In this paper we study noise sensitivity and threshold phenomena for Poisson Voronoi percolation on $$\mathbb{R}^2$$. In the setting of Boolean functions, both threshold phenomena and noise sensitivity can be understood via the study of randomized algorithms. Together with a simple discretization argument, such techniques apply also to the continuum setting. Via the study of a suitable algorithm we show that box-crossing events in Voronoi percolation are noise sensitive and present a threshold phenomenon with polynomial window. We also study the effect of other kinds of perturbations, and emphasize the fact that the techniques we use apply for a broad range of models.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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