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Exponential extinction time of the contact process on finite graphs.pdf

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Exponential extinction time of the contact process on finite graphs.pdf

Available online at ScienceDirect Stochastic Processes and their Applications 126 (2016) 1974–2013 /locate/spa Exponential extinction time of the contact process on finite graphs Thomas Mountforda, Jean-Christophe Mourrata, Daniel Valesinb,?, Qiang Yaoc a E′cole Polytechnique Fe′de′rale de Lausanne, De′partement de Mathe′matiques, 1015 Lausanne, Switzerland b University of British Columbia, Department of Mathematics, V6T1Z2 Vancouver, Canada c School of Finance and Statistics, East China Normal University, Shanghai 200241, China Received 22 August 2014; received in revised form 20 August 2015; accepted 4 January 2016 Available online 14 January 2016 Abstract We study the extinction time τ of the contact process started with full occupancy on finite trees of bounded degree. We show that, if the infection rate is larger than the critical rate for the contact process on Z, then, uniformly over all trees of degree bounded by a given number, the expectation of τ grows exponentially with the number of vertices. Additionally, for any increasing sequence of trees of bounded degree, τ divided by its expectation converges in distribution to the unitary exponential distribution. These results also hold if one considers a sequence of graphs having spanning trees with uniformly bounded degree, and provide the basis for powerful coarse-graining arguments. To demonstrate this, we consider the contact process on a random graph with vertex degrees following a power law. Improving a result of Chatterjee and Durrett (2009), we show that, for any non-zero infection rate, the extinction time for the contact process on this graph grows exponentially with the number of vertices. ?c 2016 Elsevier B.V. All rights reserved. MSC: 82C22; 05C80 Keywords: Contact process; Interacting particle systems; Metastability ? Correspondence to: University of Groningen, Johann Bernoulli Institute, Nijenborgh 9, 9747 AG Groningen, Netherlands. E-mail addresses: d.rodrigues.valesin@rug.nl, valesin@ (D. Val