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: \[\mathbb { P}_{ p,p'}(0\leftrightarrow n\mathbf { e}_1)=\psi_de^{ -\xi_{ p,p'}n}\bigl(1+o(1)\bigr)\] 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.
Categories: math.PR cond-mat.stat-mech math-ph math.MP