Subcritical percolation with a line of defects

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 $ξ_{p,p'}:=-\lim_{n\to\infty}n^{-1}\log \mathbb{P}_{p,p'}(0\leftrightarrow n\mathbf {e}_1)$ and $ξ_p:=ξ_{p,p}$. We show that there exists $p_c'=p_c'(p,d)$ such that $ξ_{p,p'}=ξ_p$ if $p'<p_c'$ and $ξ_{p,p'}<ξ_p$ 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 $ξ_p-ξ_{p,p'}$ 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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