Bernoulli Hyperplane Percolation

TL;DR

This paper investigates a dependent percolation model on high-dimensional lattices where hyperplanes are randomly removed, revealing a phase transition and different decay behaviors in the subcritical phase.

Contribution

It extends Bernoulli line percolation results to hyperplane removal, demonstrating a phase transition and decay behavior changes in the model.

Findings

Existence of a non-trivial phase transition.

Transition from exponential to power-law decay.

Model behavior differs in subcritical phase.

Abstract

We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.