  Ahlberg, Griffiths, Morris and Tassion have proved that, asymptotically
almost surely, the quenched crossing probabilities for critical planar Voronoi
percolation do not depend on the environment. We prove an analogous result for
arm events. In particular, we prove that the variance of the quenched
probability of an arm event is at most a constant times the square of the
annealed probability. The fact that the arm events are degenerate and
non-monotonic add two major difficulties. As an application, we prove that
there exists $\epsilon > 0$ such that the following holds for the annealed
percolation function $\theta^{an}$: \[ \forall p > 1/2 ,\, \theta^{an}(p) \geq
\epsilon (p-1/2)^{1-\epsilon} \, . \] One of our motivations is to provide
tools for a spectral study of Voronoi percolation.
