The Contact Process on a Finite Set. III: The Critical Case

Durrett, Richard; Schonmann, Roberto H.; Tanaka, Nelson I.

doi:10.1214/aop/1176991156

Cited by 28 publications

(17 citation statements)

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“…This result is far from easy and was proved in a number of stages ( [5], [3], Durrett and Schonmann [6], Mountford [17]). It is also known (Durrett, Schonmann and Tanaka [7]) that for λ = λ c , τ N is bounded above and below by polynomials in N .…”

Section: Introductionmentioning

confidence: 95%

The contact process on finite homogeneous trees

Stacey¹

2001

Probab Theory Relat Fields

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Let T d be a homogeneous tree in which every vertex has d +1 neighbours, where d ≥ 2. The contact process on such a tree is known to have three distinct phases. We consider the process on a finite subtree, namely the rooted tree of depth h and branching factor d, and relate the behaviour of the process on the infinite tree to its behaviour on the finite tree for large h. In the phase of strong survival, we show that with probability independent of h, the process on the subtree starting from a single infection survives for a time which is doubly exponential in h and almost exponential in the number of vertices of the finite tree. In the phase of weak survival on the infinite tree, the survival time on the finite tree is approximately linear in h. In the phase of no survival, the survival time on the finite tree is linear in h if one starts with all vertices initially infected, and bounded by a random variable (independent of h) with an exponential tail if one starts from a single infection.

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Section: Introductionmentioning

confidence: 95%

The contact process on finite homogeneous trees

Stacey¹

2001

Probab Theory Relat Fields

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“…Metastability for the contact process in this setting was also studied in [6,25]. Finally, if d = 1, it is also known that, if λ = λ c , then τ/n → ∞ and τ/n 4 → 0 in probability [14]. For the case of finite trees, the available results concerning the extinction time are contained in [10], which improves on earlier results of [26].…”

Section: Introductionmentioning

confidence: 78%

Exponential extinction time of the contact process on finite graphs

Mountford

Mourrat

Valesin

et al. 2016

Stochastic Processes and their Applications

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International audienceWe 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 [CD09], 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

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“…SinceB N 2 N &C N 2 N ; the lower bound follows from (10). For the upper bound, define We can complete the proof of Theorem 3 once we show that 1 N log PðC N Þ !…”

Section: Article In Pressmentioning

confidence: 90%

“…To verify it at l c ; note that Durrett et al [10] show that lim inf l#l c ðl À l c Þ ð1=5Þ g 2 ðlÞ 40;…”

Section: Article In Pressmentioning

confidence: 99%

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Designing a contact process: the piecewise-homogeneous process on a finite set with applications

Wagner

Anantharam

2005

Stochastic Processes and their Applications

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We consider how to choose the reproduction rates in a one-dimensional contact process on a finite set to maximize the growth rate of the extinction time with the population size. The constraints are an upper bound on the average reproduction rate, and that the rate profile must be piecewise constant. We show that the optimum growth rate is achieved by a rate profile with at most two rates, and we characterize the solution in terms of a ''spatial correlation length'' of the supercritical process. We examine the analogous problem for the simpler biased voter model, for which we completely characterize the optimum profile. The contact process proofs make use of a planar-graph duality in the graphical representation, due to Durrett and Schonmann. r 2004 Elsevier B.V. All rights reserved. MSC: primary 60K35; secondary 60K30; 82C26; 92D25; 92D30

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The Contact Process on a Finite Set. III: The Critical Case

Cited by 28 publications

References 0 publications

The contact process on finite homogeneous trees

The contact process on finite homogeneous trees

Exponential extinction time of the contact process on finite graphs

Designing a contact process: the piecewise-homogeneous process on a finite set with applications

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