Subcritical percolation with a line of defects

Abstract

We consider the Bernoulli bond percolation process ${\mathbb{P}}_{p,p^{\prime }}$ 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 $p^{\prime }$. Define \[ξ_{p,p'}:=-\lim_{n\to\infty}n^{-1}\log \mathbb{P}_{p,p'}(0\leftrightarrow n\mathbf {e}_1)\] and ${\xi }_p:={\xi }_{p,p}$. We show that there exists $p_c^{\prime }=p_c^{\prime }(p,d)$ such that ${\xi }_{p,p^{\prime }}={\xi }_p$ if $p^{\prime }<p_c^{\prime }$ and ${\xi }_{p,p^{\prime }}<{\xi }_p$ if $p^{\prime }>p_c^{\prime }$. Moreover, $p_c^{\prime }(p,2)=p_c^{\prime }(p,3)=p$, and $p_c^{\prime }(p,d)>p$ for $d\ge 4$. We also analyze the behavior of ${\xi }_p-{\xi }_{p,p^{\prime }}$ as $p^{\prime }\downarrow p_c^{\prime }$ in dimensions $d=2,3$. Finally, we prove that when $p^{\prime }>p_c^{\prime }$, 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 ${\psi }_d={\psi }_d(p,p^{\prime })$, 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.

Published in at http://dx.doi.org/10.1214/11-AOP720 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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• Localization and delocalization of random interfaces
• Marginal relevance of disorder for pinning models
• A Local Limit Theorem for random walks conditioned to stay positive
• Fluctuation theory of connectivities for subcritical random cluster models
• Equality of critical points for polymer depinning transitions with loop exponent one

Cited by in corpus (2)

• Potts models with a defect line
• Phase diagram of inhomogeneous percolation with a defect plane