IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v42y1992i1p73-90.html
   My bibliography  Save this article

Clustering in one-dimensional threshold voter models

Author

Listed:
  • Andjel, Enrique D.
  • Liggett, Thomas M.
  • Mountford, Thomas

Abstract

We consider one-dimensional spin systems in which the transition rate is 1 at site k if there are at least N sites in {k-N, k-N + 1, ..., k + N-1, k + N} at which the 'opinion' differs from that at k, and the rate is zero otherwise. We prove that clustering occurs for all N [greater-or-equal, slanted] 1 in the sense that P[[eta]t(k) [not equal to] [eta]t(j)] tends to zero as t tends to [infinity] for every initial configuration. Furthermore, the limiting distribution as t --> [infinity] exists (and is a mixture of the pointmasses on [eta] [reverse not equivalent] 1 and [eta] [reverse not equivalent] 0) if the initial distribution is translation invariant. In case N = 1, the first of these results was proved and a special case of the second was conjectured in a recent paper by Cox and Durrett. Now let D([varrho]) be the limiting density of 1's when the initial distribution is the product measure with density [rho]. If N = 1, we show that D([rho]) is concave on [0, ], convex on [, 1], and has derivative 2 at 0. If N [greater-or-equal, slanted] 2, this derivative is zero.

Suggested Citation

  • Andjel, Enrique D. & Liggett, Thomas M. & Mountford, Thomas, 1992. "Clustering in one-dimensional threshold voter models," Stochastic Processes and their Applications, Elsevier, vol. 42(1), pages 73-90, August.
  • Handle: RePEc:eee:spapps:v:42:y:1992:i:1:p:73-90
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/0304-4149(92)90027-N
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Valentina Corradi & Antonella Ianni, "undated". ""Consensus and Co-Existence in an Interactive Process of Opinion Formation''," CARESS Working Papres 98-09, University of Pennsylvania Center for Analytic Research and Economics in the Social Sciences.
    2. Xiaofeng Xue, 2015. "Asymptotic Behavior of Critical Infection Rates for Threshold-One Contact Processes on Lattices and Regular Trees," Journal of Theoretical Probability, Springer, vol. 28(4), pages 1447-1467, December.
    3. Corradi, Valentina & Ianni, Antonella, 2000. "A simple locally interactive model of ergodic and nonergodic growth," Discussion Paper Series In Economics And Econometrics 0010, Economics Division, School of Social Sciences, University of Southampton.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:42:y:1992:i:1:p:73-90. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.