Quantitative quenched Voronoi percolation and applications

TL;DR

This paper extends the understanding of quenched Voronoi percolation by analyzing arm events, showing their probabilities are environment-independent and providing bounds on variance, with implications for spectral analysis.

Contribution

It proves environment-independence of quenched arm event probabilities and bounds their variance, advancing the theoretical understanding of Voronoi percolation.

Findings

Variance of quenched arm event probability is bounded by a constant times the square of the annealed probability.

Existence of a positive epsilon such that the percolation function has a polynomial lower bound.

Environment independence of quenched crossing probabilities for arm events in Voronoi percolation.

Abstract

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.