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Synchronization via Interacting Reinforcement

Part of: Special processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  19 February 2016

Paolo Dai Pra ,
Pierre-Yves Louis  and
Ida G. Minelli
Paolo Dai Pra*
Affiliation:
Università degli Studi di Padova
Pierre-Yves Louis*
Affiliation:
Université de Poitiers
Ida G. Minelli*
Affiliation:
Università degli Studi dell'Aquila
*
∗ Postal address: Dipartimento di Matematica, Università degli Studi di Padova, via Trieste 63, I-35121 Padova, Italy. Email address: daipra@math.unipd.it.
∗∗ Postal address: Laboratoire de Mathématiques et Applications, UMR 7348 CNRS, Université de Poitiers, 11 Boulevard Marie et Pierre Curie, 86962 Technopole du Futuroscope, Chasseneuil Cedex, France. Email address: Email address: pierre.yves.louis@univ-poitiers.fr.
∗∗∗ Postal address: Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università degli Studi dell'Aquila, Via Vetoio (Coppito 1), 67100 Coppito (AQ), Italy. Email address: ida.minelli@dm.univaq.it.
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Abstract

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We consider a system of urns of Pólya type, containing balls of two colors; the reinforcement of each urn depends on both the content of the urn and the average content of all urns. We show that the urns synchronize almost surely, in the sense that the fraction of balls of a given color converges almost surely as time tends to ∞ to the same limit for all urns. A normal approximation for a large number of urns is also obtained.

Keywords

Interacting systemsynchronizationurn model

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 2 , June 2014 , pp. 556 - 568
Copyright
© Applied Probability Trust 

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