  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.
