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Subcritical percolation with a line of defects. (English) Zbl 1401.60177

Summary: We consider the Bernoulli bond percolation process \(\mathbb{P} _{p,p'}\) on the nearest-neighbor edges of \(\mathbb{Z} ^{d}\), which are open independently with probability \(p<p_{c}\), except for those lying on the first coordinate axis, for which this probability is \(p'\). Define \[ \xi_{p,p'}:=-\lim_{n\to\infty}n^{-1}\log\mathbb{P}_{p,p'}(0\leftrightarrow n\mathbf{e}_{1}) \] and \(\xi_{p}:=\xi_{p,p}\). We show that there exists \(p_{c}'=p_{c}'(p,d)\) such that \(\xi_{p,p'}=\xi_{p}\) if \(p'<p_{c}'\) and \(\xi_{p,p'}<\xi_{p}\) if \(p'>p_{c}'\). Moreover, \(p_{c}'(p,2)=p_{c}'(p,3)=p\), and \(p_{c}'(p,d)>p\) for \(d\geq 4\). We also analyze the behavior of \(\xi_{p}-\xi_{p,p'}\) as \(p'\downarrow p_{c}'\) in dimensions \(d=2,3\). Finally, we prove that when \(p'>p_{c}'\), the following purely exponential asymptotics holds: \[ \mathbb{P}_{p,p'}(0\leftrightarrow n\mathbf{e}_{1})=\psi_{d}e^{-\xi_{p,p'}n}\bigl(1+o(1)\bigr) \] for some constant \(\psi_{d}=\psi_{d}(p,p')\), uniformly for large values of \(n\). This work gives the first results on the rigorous analysis of pinning-type problems, that go beyond the effective models and don’t rely on exact computations.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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References:

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