Tightness for the interface of the one-dimensional contact process

Andjel, Enrique D.; Mountford, Thomas; Pimentel, Leandro P. R.; Valesin, Daniel

doi:10.3150/09-bej236

Cited by 12 publications

(26 citation statements)

References 10 publications

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“…(2.8) (In Lemma 2.6(i) of [2] it is shown that a vertex x satisfies certain properties in the Harris system with positive probability, and then Proposition 2.7(i) and (iii) of [2] imply that a vertex satisfying the mentioned list of properties isβ-insulating in the sense that we give here. Property (2.8) above can be obtained from Lemma 2.6(ii) of [2] through a routine argument that we will omit). The constantsβ andδ of the above proposition will be fixed throughout the paper.…”

Section: Insulating Pointsmentioning

confidence: 91%

“…In case x is β-insulating, the cone {y ∈ Z : x − βt ≤ y ≤ x + βt} is called a descendancy barrier. In [13] and [2], it was shown by the following proposition. ∀ε > 0 ∃n : ∀A ⊂ Z with #A ≥ n, P(no point of A isβ-insulating) < ε.…”

Section: Insulating Pointsmentioning

confidence: 99%

“…Throughout this paper, we fix d = 1, K ∈ N and also fix λ in the corresponding supercritical region, that is, λ > λ c (Z, K). (1.2) In [2], motivated by a conjecture in [4], the authors considered the grass-bushes-trees process with parameters given by (1.1) and (1.2) and the initial configuration in which all vertices x ≤ 0 are in state 1 and all vertices x > 0 are in state 2. For this process, defining R t = sup{x : ξ t (x) = 1} and L t = inf{x : ξ t (x) = 2}, the interval delimited by R t and L t is called the interface and |R t − L t | is the interface size (note that R t − L t is necessarily negative when the range K = 1, whereas it can be positive or negative if K > 1).…”

confidence: 99%

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Equilibrium of the interface of the grass-bushes-trees process

Andjel¹,

Mountford²,

Valesin³

2018

Bernoulli

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We consider the grass-bushes-trees process, which is a two-type contact process in which one of the types is dominant. Individuals of the dominant type can give birth on empty sites and sites occupied by nondominant individuals, whereas non-dominant individuals can only give birth at empty sites. We study the shifted version of this process so that it is 'seen from the rightmost dominant individual' (which is well defined if the process occurs in an appropriate subset of the configuration space); we call this shifted process the grass-bushes-trees interface (GBTI) process. The set of stationary distributions of the GBTI process is fully characterized, and precise conditions for convergence to these distributions are given.

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Section: Insulating Pointsmentioning

confidence: 91%

Section: Insulating Pointsmentioning

confidence: 99%

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Equilibrium of the interface of the grass-bushes-trees process

Andjel¹,

Mountford²,

Valesin³

2018

Bernoulli

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“…We remark that interface tightness is a fairly common phenomenon among many onedimensional interacting particle systems. Other models for which this has been proved include one-dimensional two-type contact processes with strong bias (in the sense that one type can overtake the other, but not vice versa) [AMPV10], or no bias (no type can infect a site occupied by the other type) [Val10,MV16], as well as one-dimensional asymmetric exclusion processes that admit so-called blocking measures (see e.g. [BM02,BLM02]).…”

Section: Introduction 1interface Tightnessmentioning

confidence: 99%

Equilibrium interfaces of biased voter models

Sun¹,

Swart²,

Yu³

2019

Ann. Appl. Probab.

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A one-dimensional interacting particle system is said to exhibit interface tightness if starting in an initial condition describing the interface between two constant configurations of different types, the process modulo translations is positive recurrent. In a biological setting, this describes two populations that do not mix, and it is believed to be a common phenomenon in one-dimensional particle systems. Interface tightness has been proved for voter models satisfying a finite second moment condition on the rates. We extend this to biased voter models. Furthermore, we show that the distribution of the equilibrium interface for the biased voter model converges to that of the voter model when the bias parameter tends to zero. A key ingredient is an identity for the expected number of boundaries in the equilibrium voter model interface, which is of independent interest. MSC 2010. Primary: 82C22; Secondary: 82C24, 82C41, 60K35.

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“…with § 2.2, p. 913 ff., Andjel et.al. [AMPV10]. The special case of the analog of Theorem 1 for the so-called basic case (i.e.…”

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confidence: 99%

The entirely coupled region of supercritical contact processes

Tzioufas

2016

J. Appl. Probab.

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We consider translation-invariant, finite range, supercritical contact processes. We show the existence of unbounded space-time cones within which the descendancy of the process from full occupancy may with positive probability be identical to that of the process from the single site at its apex. The proof comprises an argument that leans upon refinements of a successful coupling among these two processes, and is valid in d-dimensions.1. Introduction. The contact process is an extensively studied class of spatial Markov process introduced by Harris [H74] in 1974; contact distributions were considered first in Mollison [M72], for later developments in this regard, see also Mollison [M77]. The process can be viewed as a simple model for spatial growth, or the spread of an infection in a spatially structured population. In this note we will adopt the perspective and associated terminology stemming from the former interpretation. We have opted to work our proofs in detail in one (spatial) dimension, since the d-dimensional extension is directly analogous and is omitted as such. We consider the following class of translation-invariant and finite range contact processes (ξ t : t ≥ 0). Regarding sites in ξ t as occupied and others as vacant, the process with parameters µ = (µ i ; i = −M, . . . , M, i = 0) evolves according to the following local prescription: (i) Particles die at rate 1. (ii) A particle at x gives birth at rate µ y−x at y, |x − y| ≤ M, y = x. (iii) There is no more than one particle per site 1 .

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Tightness for the interface of the one-dimensional contact process

Cited by 12 publications

References 10 publications

Equilibrium of the interface of the grass-bushes-trees process

Equilibrium of the interface of the grass-bushes-trees process

Equilibrium interfaces of biased voter models

The entirely coupled region of supercritical contact processes

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