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Global divergence of spatial coalescents
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  • Published: 24 November 2010

Global divergence of spatial coalescents

  • Omer Angel1,
  • Nathanaël Berestycki2 &
  • Vlada Limic3 

Probability Theory and Related Fields volume 152, pages 625–679 (2012)Cite this article

  • 155 Accesses

  • 9 Citations

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Abstract

We study several fundamental properties of a class of stochastic processes called spatial Λ-coalescents. In these models, a number of particles perform independent random walks on some underlying graph G. In addition, particles on the same vertex merge randomly according to a given coalescing mechanism. A remarkable property of mean-field coalescent processes is that they may come down from infinity, meaning that, starting with an infinite number of particles, only a finite number remains after any positive amount of time, almost surely. We show here however that, in the spatial setting, on any infinite and bounded-degree graph, the total number of particles will always remain infinite at all times, almost surely. Moreover, if \({G\,=\,\mathbb{Z}^d}\), and the coalescing mechanism is Kingman’s coalescent, then starting with N particles at the origin, the total number of particles remaining is of order (log* N)d at any fixed positive time (where log* is the inverse tower function). At sufficiently large times the total number of particles is of order (log* N)d-2, when d > 2. We provide parallel results in the recurrent case d = 2. The spatial Beta-coalescents behave similarly, where log log N is replacing log* N.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Omer Angel

  2. Statistical Laboratory, DPMMS, University of Cambridge, Wilberforce Rd., Cambridge, CB3 0WB, UK

    Nathanaël Berestycki

  3. UMR 6632, LATP, CMI, CNRS-Université de Provence, Technopôle de Château-Gombert, 39, rue F. Joliot Curie, 13453, Marseille cedex 13, France

    Vlada Limic

Authors
  1. Omer Angel
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  2. Nathanaël Berestycki
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  3. Vlada Limic
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Corresponding author

Correspondence to Nathanaël Berestycki.

Additional information

O. Angel was supported in part by NSERC; N. Berestycki was supported in part by EPSRC grant EP/G055068/1; and V. Limic was supported in part by Alfred P. Sloan Research Fellowship, and in part by ANR MAEV grant.

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Angel, O., Berestycki, N. & Limic, V. Global divergence of spatial coalescents. Probab. Theory Relat. Fields 152, 625–679 (2012). https://doi.org/10.1007/s00440-010-0332-5

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  • Received: 24 March 2010

  • Revised: 02 November 2010

  • Published: 24 November 2010

  • Issue Date: April 2012

  • DOI: https://doi.org/10.1007/s00440-010-0332-5

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Mathematics Subject Classification (2000)

  • 60K35
  • 92D10
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