Absence of infinite cluster for critical Bernoulli percolation on slabs

TL;DR

This paper proves that at the critical point, Bernoulli percolation on certain slab graphs does not produce an infinite cluster, extending the result to symmetric finite-range models.

Contribution

It establishes the absence of infinite clusters at criticality for Bernoulli percolation on slabs and symmetric finite-range models, generalizing previous results.

Findings

No infinite cluster at criticality on slabs

Extension to finite-range models with symmetry

Applicable to graphs $bZ^2 imes bZ/kbZ$

Abstract

We prove that for Bernoulli percolation on a graph $\mathbb{Z}^2\times\{0,\dots,k\}$ ($k\ge 0$), there is no infinite cluster at criticality, almost surely. The proof extends to finite range Bernoulli percolation models on $\mathbb{Z}^2$ which are invariant under $\pi/2$-rotation and reflection.