Exponential extinction time of the contact process on finite graphs Qiang YAO Exponential extinction time of the Contents contact process on finite graphs Qiang YAO (Joint work with Thomas MOUNTFORD, Jean-Christophe MOURRAT and Daniel VALESIN) School of Finance and Statistics, East China Normal University December 7th, 2013
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction 1 Basic definitions Contents Contact process on infinite graphs Contact process on finite graphs Main results 2 Idea of proof 3 Application 4 Further work 5
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction Basic definitions Contact process on infinite graphs Contact process on finite graphs Main results Introduction Idea of proof Application Further work
History of contact processes Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction Basic definitions Contact process on infinite graphs Contact process on finite graphs First introduced by T. E. Harris (1974). Main results Idea of proof A model to describe the spread of diseases. Application Further work Two classical books: T. M. Liggett (1985). Interacting Particle Systems. T. M. Liggett (1999). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes.
Basic definitions of contact process Exponential extinction time of the contact process on finite graphs Qiang YAO G = ( V , E ): a connected undirected graph. Introduction Basic definitions The process ( ξ t : t ≥ 0): a continuous-time Markov Contact process on infinite graphs Contact process on finite process. graphs Main results State space: { A : A ⊆ V } . Idea of proof At each t , each vertex is either healthy or infected . Application ξ t is the collection of infected vertices at time t . Further work Transition rates: � ξ t → ξ t \ { x } for x ∈ ξ t at rate 1 , ξ t → ξ t ∪ { x } for x / ∈ ξ t at rate λ · |{ y ∈ ξ t : x ∼ y }| . ( ξ A t : t ≥ 0): the process with initial state A . Absorbing state: ∅ .
Contact process on infinite graphs Exponential extinction time of the contact process on finite graphs Qiang YAO We say that the process survives if the infection will Introduction Basic definitions persist with positive probability; otherwise we say Contact process on infinite graphs that it dies out . Contact process on finite graphs Main results � strong survival Idea of proof survival Application weak survival Further work Strong survival: every site will be infected infinitely many times with positive probability. Weak survival: the infection will persist with positive probability, but every site will be infected only finite times with probability 1. (Infection moves away to infinity .)
Two critical values Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction Basic definitions Contact process on infinite graphs Contact process on finite � λ 1 := inf { λ : the process survives } , graphs Main results λ 2 := inf { λ : the process survives strongly } . Idea of proof Application Further work Three phases: ξ t dies out if λ < λ 1 , ξ t survives weakly if λ 1 < λ < λ 2 , ξ t survives strongly if λ > λ 2 .
Two critical values Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction Basic definitions Contact process on infinite On integer lattices Z d ( d ≥ 1), λ 1 = λ 2 � λ c . graphs Contact process on finite graphs Main results On homogeneous trees T d ( d ≥ 3), λ 1 < λ 2 . Idea of proof Application Further work References: C. Bezuidenhout and G. R. Grimmett: The critical contact process dies out, Ann. Probab. 18 1462-1482 (1990). A. M. Stacey: The existence of an intermediate phase for the contact process on trees, Ann. Probab. 24 1711-1726 (1996).
Contact process on finite graphs Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction Basic definitions Contact process on infinite graphs Contact process on finite graphs If G is a finite graph, then the contact process on G Main results must die out. Idea of proof We are interested in the extinction time from full Application occupancy. Denote Further work τ = τ ( G ) = inf { t ≥ 0 : ξ V t = ∅} .
Contact process on finite subgraphs of Z d Exponential extinction time of the contact process on finite graphs (subcritical case) Qiang YAO Introduction Basic definitions Consider the graph { 0 , . . . , n } d (viewed as a Contact process on infinite graphs Contact process on finite subgraph of Z d ) and the distribution of τ for this graphs Main results graph, as n goes to infinity. Idea of proof If λ < λ c , then τ/ log n converges in probability to a Application constant. Further work References: R. Durrett and X. F. Liu: The contact process on a finite set, Ann. Probab. 16 1158-1173 (1988). J. W. Chen: The contact process on a finite system in higher dimensions, Chinese J. Contemp. Math. 15 13-20 (1994).
Contact process on finite subgraphs of Z d Exponential extinction time of the contact process on finite graphs (supercritical case) Qiang YAO Introduction Basic definitions Contact process on infinite If λ > λ c , then log E [ τ ] / n d converges to a positive graphs Contact process on finite graphs constant, and τ/ E [ τ ] converges in distribution to the Main results unit exponential distribution. Idea of proof Application Further work References: R. Durrett and R. H. Schonmann: The contact process on a finite set II, Ann. Probab. 16 1570-1583 (1988). T. Mountford: A metastable result for the finite multidimensional contact process, Canad. Math. Bull. 36 216-226 (1993). T. Mountford: Existence of a constant for finite system extinction, J. Statist. Phys. 96 1331-1341 (1999).
Contact process on finite subgraphs of Z d Exponential extinction time of the contact process on finite graphs (supercritical case) Qiang YAO Introduction Basic definitions Contact process on infinite graphs Contact process on finite graphs Main results In the supercritical case, the order of magnitude of Idea of proof the extinction time is exponential in the number of Application vertices of the graph; the process is said to exhibit Further work metastability , meaning that it persists for a long time in a state that resembles an equilibrium and then quickly moves to its true equilibrium ( ∅ in this case).
Contact process on finite subgraphs of Z d Exponential extinction time of the contact process on finite graphs (critical case) Qiang YAO Introduction Basic definitions Contact process on infinite graphs Contact process on finite graphs If d = 1 and λ = λ c , then τ/ n → ∞ and τ/ n 4 → 0 Main results in probability. Idea of proof Application Further work Reference: R. Durrett, R. H. Schonmann and N. Tanaka: The contact process on a finite set III. The critical case, Ann. Probab. 17 1303-1321 (1989).
Contact process on finite subgraphs of T d Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction Basic definitions Contact process on infinite graphs Contact process on finite graphs Fix d ≥ 2, let T h d be the finite subgraph of T d Main results defined by considering up to h generations from the Idea of proof root and again take the contact process started from Application full occupancy on this graph, with associated Further work extinction time τ . If λ < λ 2 , then there exist constants c , C > 0 such that P ( ch ≤ τ ≤ Ch ) → 1 as h → ∞ .
Contact process on finite subgraphs of T d Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction Basic definitions Contact process on infinite graphs If λ > λ 2 , then for any σ < 1 there exist c 1 , c 2 > 0 Contact process on finite graphs � τ > c 1 e c 2 ( σ d ) h � such that P → 1 as h → ∞ . Main results Idea of proof This implies that τ is at least as large as a stretched Application exponential function of the number of vertices, Further work ( d + 1) h . Reference: A. M. Stacey: The contact process on finite homogeneous trees, Probab. Th. Rel. Fields 121 551-576 (2001).
Contact process on general finite graphs Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction Basic definitions Contact process on infinite graphs Contact process on finite graphs Main results Idea of proof As far as we know, no rigorous results are available Application concerning finite graphs which are not regular. Further work
Exponential extinction time of the contact process on finite graphs Qiang YAO Introduction Basic definitions Contact process on infinite graphs Contact process on finite graphs Main results Main results Idea of proof Application Further work
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