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Absence of infinite cluster for critical Bernoulli percolation on slabs. (English) Zbl 1342.82076
Summary: We prove that for Bernoulli percolation on a graph \(\mathbb{Z}^2\times\{0,\dots,k\}\) \((k\geq0)\), there is no infinite cluster at criticality, almost surely. The proof extends to finite-range Bernoulli percolation models on \(\mathbb{Z}^2\) that are invariant under \({\frac{\pi}2}\)-rotation and reflection.

MSC:
82B43 Percolation
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B26 Phase transitions (general) in equilibrium statistical mechanics
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