If we consider the contact process with infection rate λ on a random
graph on n vertices with power law degree distributions, mean field
calculations suggest that the critical value λc of the infection rate
is positive if the power α>3. Physicists seem to regard this as an
established fact, since the result has recently been generalized to bipartite
graphs by G\'{o}mez-Garde\~{n}es et al. [Proc. Natl. Acad. Sci. USA 105 (2008)
1399--1404]. Here, we show that the critical value λc is zero for any
value of α>3, and the contact process starting from all vertices
infected, with a probability tending to 1 as n→∞, maintains a positive
density of infected sites for time at least exp(n1−δ) for any
δ>0. Using the last result, together with the contact process duality,
we can establish the existence of a quasi-stationary distribution in which a
randomly chosen vertex is occupied with probability ρ(λ). It is
expected that ρ(λ)∼Cλβ as λ→0. Here we
show that α−1≤β≤2α−3, and so β>2 for α>3. Thus
even though the graph is locally tree-like, β does not take the mean
field critical value β=1.Comment: Published in at http://dx.doi.org/10.1214/09-AOP471 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org