The Annals of Probability

Subcritical percolation with a line of defects

S. Friedli, D. Ioffe, and Y. Velenik

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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.

Article information

Ann. Probab., Volume 41, Number 3B (2013), 2013-2046.

First available in Project Euclid: 15 May 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Percolation local limit theorem renewal Russo formula pinning random walk correlation length Ornstein–Zernike analyticity


Friedli, S.; Ioffe, D.; Velenik, Y. Subcritical percolation with a line of defects. Ann. Probab. 41 (2013), no. 3B, 2013--2046. doi:10.1214/11-AOP720.

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