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Subcritical percolation with a line of defects

We consider the Bernoulli bond percolation process $\mathbb{P}_{p,p&#39;}$ 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 $p&#39;$. Define \[ξ_{p,p&#39;}:=-\lim_{n\to\infty}n^{-1}\log \mathbb{P}_{p,p&#39;}(0\leftrightarrow n\mathbf {e}_1)\] and $ξ_p:=ξ_{p,p}$. We show that there exists $p_c&#39;=p_c&#39;(p,d)$ such that $ξ_{p,p&#39;}=ξ_p$ if $p&#39;<p_c&#39;$ and $ξ_{p,p&#39;}<ξ_p$ if $p&#39;>p_c&#39;$. Moreover, $p_c&#39;(p,2)=p_c&#39;(p,3)=p$, and $p_c&#39;(p,d)>p$ for $d\geq 4$. We also analyze the behavior of $ξ_p-ξ_{p,p&#39;}$ as $p&#39;\downarrow p_c&#39;$ in dimensions $d=2,3$. Finally, we prove that when $p&#39;>p_c&#39;$, the following purely exponential asymptotics holds: \[\mathbb {P}_{p,p&#39;}(0\leftrightarrow n\mathbf {e}_1)=ψ_de^{-ξ_{p,p&#39;}n}\bigl(1+o(1)\bigr)\] for some constant $ψ_d=ψ_d(p,p&#39;)$, 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&#39;t rely on exact computations.