Contact process on one-dimensional long range percolation (Q907975)

This short paper is devoted for the study of the contact process on one-dimensional long range percolation graph \(G_s\) with exponent \(s > 1\). This graph is defined as follows: independently for any \(i\) and \(j\) in \(\mathbb Z\) there is an edge connecting them with probability \(|i-j|^{-s}\). On this graph a contact process can be defined as a Markov process \((\xi_t)_{t\geq 0}\) on \(\{{0,1}^V\}\) with \(\lambda>0\) infection rate, where \(V\) is a set of graph vertices that can be infected (state 1) or healthy (state 0). Taking into account both above issues and results presented in [\textit{L. Ménard} and \textit{A. Singh}, ``Percolation by cumulative merging and phase transition for the contact process on random graphs'', \url{arXiv:1502.06982}], the author was able to show \(G_s\) can be seen as the gluing of i.i.d. finite subgraphs. Studying the moment of the total weight of a subgraph, he was able to prove this main theorem.