Results on the contact process with dynamic edges or under renewals

Hilário, Marcelo R.; Ungaretti, Daniel; Valesin, Daniel; Vares, Maria Eulália

doi:10.48550/arxiv.2108.03219

Cited by 3 publications

(7 citation statements)

References 22 publications

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“…With this we showed that in particular on subexponential growth graphs, as for example Z d , the results shown in [LR20] and [Hil+21] regarding the CPDP do not depend on their stationarity assumption of the background but hold in general.…”

Section: Discussionmentioning

confidence: 56%

“…Furthermore, they studied the asymptotic behavior for survival as the update speed of an edge α + β tends to 0 or ∞. Note that just recently [Hil+21] provide some further results in this direction for the CPDP on Z d . In all of these results the background is assumed to be stationary, i.e the initial distribution is the unique invariant law.…”

Section: Discussionmentioning

confidence: 99%

“…One of the first to explicitly study a contact process with dynamical rates was Broman [Bro07]. Other works have been for example [SW08], [LR20] and [Hil+21]. A related variation are multi type contact process see for example [DS91], [DM91], [Rem08] and [Kuo16].…”

Section: Introductionmentioning

confidence: 99%

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Contact process in an evolving random environment

Seiler¹,

Sturm²

2022

Preprint

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In this paper we introduce a contact process in an evolving random environment (CPERE) on a connected and transitive graph with bounded degree, where we assume that this environment is described through an ergodic spin systems with finite range. We show that under a certain growth condition the phase transition of survival is independent of the initial configuration of the process. We study the invariant laws of the CPERE and show that under aforementioned growth condition the phase transition for survival coincides with the phase transition of non-triviality of the upper invariant law. Furthermore, we prove continuity properties for the survival probability and derive equivalent conditions for complete convergence, in an analogous way as for the classical contact process. We then focus on the special case, where the evolving random environment is described through a dynamical percolation. We show that the contact process on a dynamical percolation on the d-dimensional integer dies out a.s. at criticality and complete convergence holds for all parameter choices. In the end we derive some comparison results between a dynamical percolation and ergodic spin systems with finite range such that we get bounds on the survival probability of a contact process in an evolving random environment and we determine in this case that complete convergence holds in a certain parameter regime.

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Section: Discussionmentioning

confidence: 56%

Section: Discussionmentioning

confidence: 99%

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Contact process in an evolving random environment

Seiler¹,

Sturm²

2022

Preprint

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“…The aim of this project is to investigate how temporal variability in the surrounding medium can change the qualitative behaviour of diffusion processes in that medium using the contact process as an example. Similar problems have been studied recently for regular graphs, for example in the work of da Silva et al [19] and Hilário et al [9], but our focus is on scale-free and hence irregular graphs, which feature very different behaviour. We interpolate between two extreme scenarios, on the one hand the infection on the static, and hence infinitely slowly evolving, scale-free network, on the other hand the mean-field model, which effectively corresponds to an infinitely fast network evolution.…”

Section: Introductionmentioning

confidence: 93%

The contact process on dynamical scale-free networks

Jacob¹,

Linker²,

Mörters³

2022

Preprint

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We investigate the contact process on four different types of scale-free inhomogeneous random graphs evolving according to a stationary dynamics, where each potential edge is updated with a rate depending on the strength of the adjacent vertices. Depending on the type of graph, the tail exponent of the degree distribution and the updating rate, we find parameter regimes of fast and slow extinction and in the latter case identify metastable exponents that undergo first order phase transitions.Résumé: Nous étudions le processus de contact sur quatre types différents de graphes aléatoires inhomogènes invariants d'échelle évoluant selon une dynamique stationnaire, où chaque arête potentielle est rafraîchie à un taux dépendant de la force des sommets adjacents. En fonction du type de graphe, de l'exposant de la queue de distribution des degrés et du taux de rafraîchissement, nous trouvons des régimes d'extinction rapide ou lente et, dans ce dernier cas, nous identifions des exposants métastables qui subissent des transitions de phase de premier ordre.

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“…Proposition 1.3.1 and Theorem 1.3.2 (i) show that for p small enough λ c (v, p) → ∞ as v → 0. But recently Hilário et al [Hil+21] have studied a robust renormalization approach for generalized contact process. They call any process that is obtained from a percolative structure of recovery and transmission marks in same way as the contact process, but the distribution of these marks is given through some other Poisson point.…”

Section: Chapter 1 Introduction and Main Resultsmentioning

confidence: 99%

The contact process in an evolving random environment

Marco¹

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Results on the contact process with dynamic edges or under renewals

Cited by 3 publications

References 22 publications

Contact process in an evolving random environment

Contact process in an evolving random environment

The contact process on dynamical scale-free networks

The contact process in an evolving random environment

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