• Jeon, Intae (Department of Mathematics, The Catholic University of Korea)
  • Received : 2013.06.24
  • Accepted : 2013.10.31
  • Published : 2013.12.25


We consider zero range processes with density dependent jump rates g given by $g=g(n,k)=g_1(n)g_2(k/n)$ with $g_1(x)=x^{-\alpha}$ and $$g_2(x)=\{^{x^{-\alpha}\;if\;a&lt;x}_{Mx^{-\alpha}\;if\;x{\leq}a}$$. (0.1) In this case, with 1/2 < a < 1 and ${\alpha}$ > 0, we show that non-complete condensation occurs with maximum cluster size an. More precisely, for any ${\epsilon}$ > 0, there exists $M^*$ > 0 such that, for any 0 < M ${\leq}M^*$, the maximum cluster size is between (a - ${\epsilon}$)n and (a + ${\epsilon}$)n for large n. This provides a simple example of non-complete condensation under perturbation of rates which are deep in the range of perfect condensation (e.g. ${\alpha}$ >> 1) and supports the instability of the condensation transition.


  1. I. Armendariz and M. Loulakis, Thermodynamic Limit for the Invariant Measures in Supercritical Zero Range Processes, Prob. Th. Rel. Fields, 145 (2009), no.1-2, 175-188.
  2. J. Beltran and C. Landim, Metastability of reversible condensed zero range processes on a finite set, Prob. Th. Rel. Fields, 152 (2012), no. 3, 781-807.
  3. L. Molino, P. Chleboun, and S. Grosskinsky, Condensation in randomly perturbed zero-range processes, J. Phys. A: Math. Theor., 45 (2012), no 20, 205001.
  4. M. R. Evans, Phase transitions in one-dimensional nonequilibrium systems, Braz. J. Phys. 30 (2000), 42-57.
  5. M. R. Evans and T. Hanney, Nonequilibrium statistical mechanics of the zero-range process and related models, J. Phys. A: Math. Gen., 38 (2005), 195-240.
  6. S. Grosskinsky and G. Schutz, Discontinuous condensation transition and nonequivalence of ensembles in a zero-range process, J. Stat. Phys., 132 (2008), no. 1, 77-108.
  7. S. Grosskinsky, P. Chleboun, and G. Schutz, Instability of condensation in the zero-range process with random interaction, Phys. Rev. E: Stat. Nonlin. Soft Matter Phys., 78 (2008), 030101.
  8. I. Jeon, Existence of gelling solutions for coagulation fragmentation equations, Comm. Math. Phys., 194 (1998), 541-567.
  9. I. Jeon, Phase transition for perfect condensation and instability under the perturbations on jump rates of the Zero Range Process, J. Phys. A: Math. and Theor., 43 (2010), 235002.
  10. I. Jeon, Condensation in perturbed zero range processes, J. Phys. A: Math. and Theor., 44 (2011), 255002.
  11. I. Jeon and P. March, Condensation transition for zero range invariant measures, In Stochastic models. Proceedings of the International Conference on Stochastic Models in Honor of Professor Donald A. Dawson (Luis G. Gorostiza, B. Gail Ivanoff, eds.), 2000, 233-244.
  12. I. Jeon, P. March, and B. Pittel, Size of the largest cluster under zero-range invariant measures, Ann. of Prob., 28 (2000), 1162-1194.
  13. T. M. Liggett, Interacting Particle Systems, Springer-Verlag, 1985.
  14. F. Spitzer, Interaction of Markov processes, Adv. Math., 5 (1970), 246-290.