Metastability for the contact process on the preferential attachment graph

We consider the contact process on the preferential attachment graph. The work of Berger, Borgs, Chayes and Saberi confirmed physicists’ predictions that the contact process starting from a typical vertex becomes epidemic for an arbitrarily small infection rate \(\lambda\) with positive probability. More precisely, they showed that with probability \(\lambda^{\Theta (1)}\), it survives for a time exponential in the largest degree. Here we obtain sharp bounds for the density of infected sites at a time close to exponential in the number of vertices (up to some logarithmic factor). In addition, a metastable result for the extinction time is also proved.