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arXiv · 1103.0411

Subcritical percolation with a line of defects

S. Friedli·D. Ioffe·Y. Velenik
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Abstract

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 $p p_c'$. 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 $ξ_p-ξ_{p,p'}$ as $p'\downarrow p_c'$ in dimensions $d=2,3$. Finally, we prove that when $p'>p_c'$, the following purely exponential asymptotics holds: \[\mathbb {P}_{p,p'}(0\leftrightarrow n\mathbf {e}_1)=ψ_de^{-ξ_{p,p'}n}\bigl(1+o(1)\bigr)\] for some constant $ψ_d=ψ_d(p,p')$, 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.

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S. Friedli, D. Ioffe, Y. Velenik. 2013-07-08. Subcritical percolation with a line of defects. https://doi.org/10.1214/11-aop720

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