Quantitative quenched Voronoi percolation and applications

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.