Absence of infinite cluster for critical Bernoulli percolation on slabs

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.

17 pages, 5 figures

References in corpus (2)

• Random Currents and Continuity of Ising Model's Spontaneous Magnetization
• Absence of site percolation at criticality in Z^2 x {0,1}

Cited by in corpus (2)

• Locality of percolation critical probabilities: uniformly nonamenable case
• The phase transitions of the random-cluster and Potts models on slabs with $q\ge 1$ are sharp