Open Access
Translator Disclaimer
May 2013 Subcritical percolation with a line of defects
S. Friedli, D. Ioffe, Y. Velenik
Ann. Probab. 41(3B): 2013-2046 (May 2013). DOI: 10.1214/11-AOP720


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.


Download Citation

S. Friedli. D. Ioffe. Y. Velenik. "Subcritical percolation with a line of defects." Ann. Probab. 41 (3B) 2013 - 2046, May 2013.


Published: May 2013
First available in Project Euclid: 15 May 2013

zbMATH: 06216095
MathSciNet: MR3098065
Digital Object Identifier: 10.1214/11-AOP720

Primary: 60K35 , 82B43

Keywords: analyticity , correlation length , local limit theorem , Ornstein–Zernike , percolation , Pinning , Random walk , Renewal , Russo formula

Rights: Copyright © 2013 Institute of Mathematical Statistics


Vol.41 • No. 3B • May 2013
Back to Top