Subcritical percolation with a line of defects

TL;DR

This paper investigates how a line of defects affects the decay rate of connectivity probabilities in subcritical percolation on dimensional lattices, revealing phase transitions and exponential asymptotics.

Contribution

It provides the first rigorous analysis of pinning phenomena in percolation models beyond effective theories and exact calculations.

Findings

Existence of a critical defect probability p'_c where decay rates change

For dimensions 2 and 3, p'_c equals the base percolation threshold p

For dimensions 4 and higher, p'_c exceeds p

Abstract

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.