Sharp phase transition for Cox percolation

TL;DR

This paper establishes the precise point at which a Cox percolation model transitions from non-percolating to percolating, using advanced probabilistic inequalities and coarse-graining techniques.

Contribution

It proves the sharpness of the phase transition for Cox percolation models under finite dependence and local boundedness, without requiring the FKG inequality.

Findings

Proved sharp phase transition for Cox percolation models.

Used OSSS inequality and coarse-graining in the proof.

Applicable to models with finite dependence and local boundedness.

Abstract

We prove the sharpness of the percolation phase transition for a class of Cox percolation models, i.e., models of continuum percolation in a random environment. The key requirements are that the environment has a finite range of dependence and satisfies a local boundedness condition, however the FKG inequality need not hold. The proof combines the OSSS inequality with a coarse-graining construction.