# Abstract

Subcritical Percolation with a Line of Defects S. Friedli, D. Ioffe, Y. Velenik Ann. Probab. 41, 2013-2046 (2013). 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 $p<p_{\rm c}$, 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, $$\mathbb{P}_{p,p'}(0\leftrightarrow n\mathbf{e}_1) = \psi_d\, e^{-\xi_{p,p'} n} \, (1+o(1)),$$ 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. Key words: Percolation, local limit theorem, renewal, Russo formula, pinning, random walk, correlation length, Ornstein-Zernike, analyticity.