1103.0411 (S. Friedli et al.)

Subcritical percolation with a line of defects    [PDF]

S. Friedli, D. Ioffe, Y. Velenik

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 $pp_c'$. 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.

View original: http://arxiv.org/abs/1103.0411