The Annals of Applied Probability

Rate of relaxation for a mean-field zero-range process

Benjamin T. Graham

Full-text: Open access


We study the zero-range process on the complete graph. It is a Markov chain model for a microcanonical ensemble. We prove that the process converges to a fluid limit. The fluid limit rapidly relaxes to the appropriate Gibbs distribution.

Article information

Ann. Appl. Probab., Volume 19, Number 2 (2009), 497-520.

First available in Project Euclid: 7 May 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C20: Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs

Mean field zero-range process balls boxes Markov chain relaxation spectral gap log Sobolev


Graham, Benjamin T. Rate of relaxation for a mean-field zero-range process. Ann. Appl. Probab. 19 (2009), no. 2, 497--520. doi:10.1214/08-AAP549.

Export citation


  • [1] Csiszár, I. (1967). Information-type measures of difference of probability distributions and indirect observations. Studia Sci. Math. Hungar. 2 299–318.
  • [2] Darling, R. W. R. and Norris, J. R. (2005). Structure of large random hypergraphs. Ann. Appl. Probab. 15 125–152.
  • [3] Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750.
  • [4] Ehrenfest, P. and Ehrenfest, T. (1907). Über zwei bekannte Einwände gegen das Boltzmannsche H-theorem. Phys. Zeit. 8 311–314.
  • [5] Ehrenfest, P. and Ehrenfest, T. (1959). The Conceptual Foundations of the Statistical Approach in Mechanics. Cornell Univ. Press, Ithaca, NY. Translated by M. J. Moravcsik.
  • [6] Feller, W. (1968). An Introduction to Probability Theory and Its Applications. Vol. I. 3rd ed. Wiley, New York.
  • [7] Goldschmidt, C. A. (2003). Large random hypergraphs. Ph.D. thesis, Univ. Cambridge. Available at
  • [8] Graham, B. T. (2007). Interacting stochastic systems. Ph.D. thesis, Univ. Cambridge. Available at
  • [9] Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. Wiley, New York.
  • [10] Kahn, J., Kalai, G. and Linial, N. (1988). The influence of variables on Boolean functions. In Proceedings of 29th Symposium on the Foundations of Computer Science 68–80. Computer Science Press.
  • [11] Lindvall, T. (2002). Lectures on the Coupling Method. Dover, Mineola, NY. Corrected reprint of the 1992 original.
  • [12] McCoy, B. M. and Wu, T. T. (1973). The Two-Dimensional Ising Model. Harvard Univ. Press, Cambridge.
  • [13] McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics. Algorithms Combin. 16 195–248. Springer, Berlin.
  • [14] Miclo, L. (1999). An example of application of discrete Hardy’s inequalities. Markov Process. Related Fields 5 319–330.
  • [15] Mitzenmacher, M. (1999). On the analysis of randomized load balancing schemes. Theory Comput. Syst. 32 361–386. ACM Symposium on Parallel Algorithms and Architectures (Padua, 1996).
  • [16] Morris, B. (2006). Spectral gap for the zero range process with constant rate. Ann. Probab. 34 1645–1664.
  • [17] Peres, Y. (2005). Mixing for Markov chains and spin systems. Lecture Notes 2005 UBC Summer School. Available at