The extinction of the contact process in a one-dimensional random environment with long-range interactions

We study the contact process on the long-range percolation cluster on $\mathbb{Z}$ where each edge $⟨i,j⟩$ is open with probability $\mathrm{\mid }i-j{\mathrm{\mid }}^{-s}$ for $s>2$. Using a renormalization procedure we apply Peierls-type argument to prove that the contact process dies out if the transmission rate is smaller than a critical threshold. Our methods involve the control of crossing probabilities for percolation on randomly-stretched lattices as in https://doi.org/10.1214/22-AAP1887.