Unusual properties of contact processes on percolated graphs

In this paper we will consider the contact process in a very simple type of random environment that physicists call the random dilution model. We start with the contact process on a graph, here either ${\mathbb{Z}}^d$, a $d$-dimensional torus or an \ER graph, and then flip independent $(1-p)$ coins to delete edges, or delete vertices. Let $p^{\ast }$ be the threshold for percolation in the diluted graph. We will primarily be concerned with two phenomena. (i) The critical value for the contact process on the dliuted graph ${\lambda }_c(p)$ does not converge to $\mathrm{\infty }$ as $p\downarrow p^{\ast }$. (ii) In contrast to the contact process on a homogeneous graph, the density of 1’s starting from all sites occupied converges to 0 at a polynomial rate when $p<p^{\ast }$ (the ``Griffiths phase’') and like $c\mathrm{/}(\log t{)}^a$ when $p=p^{\ast }$.