Math @ Duke

Publications [#243515] of Richard T. Durrett
Papers Published
 Chatterjee, S; Durrett, R, Contact processes on random graphs with power law degree distributions have critical value 0,
The Annals of Probability, vol. 37 no. 6
(November, 2009),
pp. 23322356, ISSN 00911798 [doi]
(last updated on 2018/10/14)
Abstract: 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 GómezGardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 13991404]. 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(n1δ) for any δ>0. Using the last result, together with the contact process duality, we can establish the existence of a quasistationary 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 treelike, β does not take the mean field critical value β = 1. © Institute of Mathematical Statistics, 2009.


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