Electronic Communications in Probability

Propagation of chaos for a balls into bins model

Nicoletta Cancrini and Gustavo Posta

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Abstract

Consider a finite number of balls initially placed in $L$ bins. At each time step a ball is taken from each non-empty bin. Then all the balls are uniformly reassigned into bins. This finite Markov chain is called Repeated Balls-into-Bins process and is a discrete time interacting particle system with parallel updating. We prove that, starting from a suitable (chaotic) set of initial states, as $L\to +\infty $, the numbers of balls in each bin become independent from the rest of the system i.e. we have propagation of chaos. We furthermore study some equilibrium properties of the limiting nonlinear process.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 1, 9 pp.

Dates
Received: 21 September 2018
Accepted: 18 December 2018
First available in Project Euclid: 4 January 2019

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1546571102

Digital Object Identifier
doi:10.1214/18-ECP204

Zentralblatt MATH identifier
07023492

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60B10: Convergence of probability measures

Keywords
chaos propagation interacting particle system parallel updates queues network

Rights
Creative Commons Attribution 4.0 International License.

Citation

Cancrini, Nicoletta; Posta, Gustavo. Propagation of chaos for a balls into bins model. Electron. Commun. Probab. 24 (2019), paper no. 1, 9 pp. doi:10.1214/18-ECP204. https://projecteuclid.org/euclid.ecp/1546571102


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References

  • [1] L. Andreis, P. Dai Pra, M. Fischer: McKean-Vlasov limit for interacting systems with simultaneous jumps, arXiv:1704.01052.
  • [2] L. Becchetti, A. Clementi, E. Natale, F. Pasquale, G. Posta: Self-stabilizing repeated balls-into-bins. Distrib. Comput. (2017), 1–10.
  • [3] P. Billingsley: Probability and measure (third edition). John Wiley & Sons, New York, 1995.
  • [4] B. Fristedt, L. Gray: A modern approach to probability theory. Birkhäuser, Boston, 1997.
  • [5] J. R. Jackson: Jobshop-like queueing systems. Management Sciences Research Project 81 (1963).
  • [6] M. Kac: Foundations of kinetic theory. Proceedings of the Third Berkley Symposium on Mathematical Statistics and Probability 1954–1955 III, University of California Press, Berkley and Los Angeles, 1956, 171–197.
  • [7] F. P. Kelly: Networks of queues. Advances in Appl. Probability 8 (1976), no. 2, 416–432.
  • [8] A. K. Erlang: Sandsynlighedsregning Og Telefonsamtaler. Nyt Tidsskrift for Matematik 20 (1909), 33–39.
  • [9] K. Nakagawa: On the series expansion of the stationary probabilities of an M/D/1 queue, J. Oper. Res. Soc. Japan 48 (2005) no. 2, 111–122.
  • [10] F. Spitzer: Interaction of Markov processes. Advances in Math. 5 (1970), 246–290.
  • [11] A-S. Sznitman: Topics in propagation of chaos. École d’Été de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Mathematics 1464, Springer, Berlin, 1991, 165–251.