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arxiv: 1401.7130 · v1 · pith:PD4J6AWNnew · submitted 2014-01-28 · 🧮 math.PR · math-ph· math.MP

Absence of infinite cluster for critical Bernoulli percolation on slabs

classification 🧮 math.PR math-phmath.MP
keywords bernoullipercolationclusterinfinitemathbbabsencealmostcritical
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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.

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