Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Supercritical behavior of asymmetric zero-range process with sitewise disorder

C. Bahadoran, T. Mountford, K. Ravishankar, and E. Saada

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Abstract

We establish necessary and sufficient conditions for weak convergence to the upper invariant measure for one-dimensional asymmetric nearest-neighbour zero-range processes with non-homogeneous jump rates. The class of “environments” considered is close to that considered by (Stochastic Process. Appl. 90 (2000) 67–81), while our class of processes is broader. We also give in arbitrary dimension a simpler proof of the result of (In Asymptotics: Particles, Processes and Inverse Problems (2007) 108–120 Inst. Math. Statist.) with weaker assumptions.

Résumé

Nous établissons des conditions nécessaires et suffisantes de convergence faible vers la mesure invariante maximale pour le processus de zero-range asymétrique à plus proche voisin en dimension un, avec des taux de sauts inhomogènes. La classe d’« environnements » considérée est proche de celle considérée dans (Stochastic Process. Appl. 90 (2000) 67–81), mais la classe de processus concernée est plus large. Nous donnons également, en dimension quelconque, une preuve plus simple du résultat de (In Asymptotics: Particles, Processes and Inverse Problems (2007) 108–120 Inst. Math. Statist.) sous des hypothèses plus faibles.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 2 (2017), 766-801.

Dates
Received: 18 November 2014
Revised: 19 November 2015
Accepted: 13 December 2015
First available in Project Euclid: 11 April 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1491897745

Digital Object Identifier
doi:10.1214/15-AIHP736

Mathematical Reviews number (MathSciNet)
MR3634274

Zentralblatt MATH identifier
1369.60065

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments
Secondary: 82C22: Interacting particle systems [See also 60K35]

Keywords
Zero-range process Site disorder Supercritical initial condition Large-time convergence Critical invariant measure Escape of mass Hydrodynamic limit

Citation

Bahadoran, C.; Mountford, T.; Ravishankar, K.; Saada, E. Supercritical behavior of asymmetric zero-range process with sitewise disorder. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 2, 766--801. doi:10.1214/15-AIHP736. https://projecteuclid.org/euclid.aihp/1491897745


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References

  • [1] E. Andjel, P. A. Ferrari, H. Guiol and C. Landim. Convergence to the maximal invariant measure for a zero-range process with random rates. Stochastic Process. Appl. 90 (2000) 67–81.
  • [2] E. D. Andjel. Invariant measures for the zero-range process. Ann. Probab. 10 (1982) 525–547.
  • [3] C. Bahadoran and T. Bodineau. Existence of a plateau for the flux function of TASEP with site disorder. Available at arXiv:1602.06718.
  • [4] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. A constructive approach to Euler hydrodynamics for attractive processes. Application to $k$-step exclusion. Stochastic Process. Appl. 99 (1) (2002) 1–30.
  • [5] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34 (4) (2006) 1339–1369.
  • [6] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Euler hydrodynamics for attractive particle systems in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 50 (2) (2014) 403–424.
  • [7] C. Bahadoran and T. S. Mountford. Convergence and local equilibrium for the one-dimensional nonzero mean exclusion process. Probab. Theory Related Fields 136 (3) (2006) 341–362.
  • [8] C. Bahadoran, T. S. Mountford, K. Ravishankar and E. Saada. Supercriticality conditions for the asymmetric zero-range process with sitewise disorder. Braz. J. Probab. Stat. 29 (2) (2015) 313–335.
  • [9] M. Balázs and T. Seppäläinen. A convexity property of expectations under exponential weights. Preprint, 2007. Available at arXiv.org/abs/0707.4273.
  • [10] M. Balázs, F. Rassoul-Agha, T. Seppäläinen and S. Sethuraman. Existence of the zero range process and a deposition model with superlinear growth rates. Ann. Probab. 35 (4) (2007) 1201–1249.
  • [11] I. Benjamini, P. A. Ferrari and C. Landim. Asymmetric conservative processes with random rates. Stochastic Process. Appl. 61 (2) (1996) 181–204.
  • [12] M. Bramson and T. M. Liggett. Exclusion processes in higher dimensions: Stationary measures and convergence. Ann. Probab. 33 (2005) 2255–2313.
  • [13] M. Bramson and T. Mountford. Stationary blocking measures for one-dimensional nonzero mean exclusion processes. Ann. Probab. 30 (3) (2002) 1082–1130.
  • [14] P. Chleboun and S. Grosskinsky. Condensation in stochastic particle systems with stationary product measures. J. Stat. Phys. 54 (2014) 432–465.
  • [15] C. Cocozza-Thivent. Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70 (4) (1985) 509–523.
  • [16] M. Ekhaus and L. Gray. A strong law for the motion of interfaces in particle systems. Unpublished manuscript, 1994.
  • [17] M. R. Evans. Bose–Einstein condensation in disordered exclusion models and relation to traffic flow. Europhys. Lett. 36 (1) (1996) 13. DOI: 10.1209/epl/i1996-00180-y.
  • [18] P. A. Ferrari and V. V. Sisko. Escape of mass in zero-range processes with random rates. In Asymptotics: Particles, Processes and Inverse Problems 108–120. IMS Lecture Notes Monogr. Ser. 55. Inst. Math. Statist, Beachwood, OH, 2007.
  • [19] C. Godrèche and J. M. Luck. Condensation in the inhomogeneous zero-range process: An interplay between interaction and diffusion disorder. J. Stat. Mech. Theory Exp. (2012) P12013.
  • [20] I. Grigorescu, M. Kang and T. Seppäläinen. Behavior dominated by slow particles in a disordered asymmetric exclusion process. Ann. Appl. Probab. 14 (2004) 1577–1602.
  • [21] T. E. Harris. Nearest-neighbor Markov interaction processes on multidimensional lattices. Adv. in Math. 9 (1972) 66–89.
  • [22] K. Jain and M. Barma. Dynamics of a disordered, driven zero-range process in one dimension. Phys. Rev. Lett. 91 (2003) 135701.
  • [23] A. Janowski and J. L. Lebowitz. Finite-size effects and shock fluctuations in the asymmetric simple-exclusion process. Phys. Rev. A 45 (1992) 618–625.
  • [24] J. Krug. Phase separation in disordered exclusion models. Braz. J. Phys. 30 (2000) 97–104.
  • [25] J. Krug and T. Seppäläinen. Hydrodynamics and platoon formation for a totally asymmetric exclusion process with particlewise disorder. J. Stat. Phys. 95 (1999) 525–567.
  • [26] C. Landim. Conservation of local equilibrium for attractive particle systems on $\mathbb{Z}^{d}$. Ann. Probab. 21 (4) (1993) 1782–1808.
  • [27] C. Landim. Hydrodynamical limit for space inhomogeneous one-dimensional totally asymmetric zero-range processes. Ann. Probab. 24 (1996) 599–638.
  • [28] C. Landim. Metastability for a non-reversible dynamics: The evolution of the condensate in totally asymmetric zero range processes. Comm. Math. Phys. 330 (2014) 1–32.
  • [29] T. M. Liggett. Coupling the simple exclusion process. Ann. Probab. 4 (3) (1976) 339–356.
  • [30] T. M. Liggett. Interacting Particle Systems. Classics in Mathematics. Springer, Berlin, 2005. Reprint of the 1985 original.
  • [31] E. Pardoux. Processus de Markov et Applications. Dunod, Paris, 2007.
  • [32] F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on $\mathbb{Z}^{d}$. Comm. Math. Phys. 140 (3) (1991) 417–448.
  • [33] F. Rezakhanlou. Continuum limit for some growth models. II. Ann. Probab. 29 (3) (2001) 1329–1372.
  • [34] R. T. Rockafellar. Convex Analysis. Princeton University Press, Princeton, 1970.
  • [35] T. Seppäläinen. Existence of hydrodynamics for the totally asymmetric simple $K$-exclusion process. Ann. Probab. 27 (1) (1999) 361–415.
  • [36] J. M. Swart A course in interacting particle systems. Preprint, 2015. Available at http://staff.utia.cas.cz/swart/lecture_notes/partic15_2.pdf.
  • [37] G. Tripathy and M. Barma. Driven lattice gases with quenched disorder: Exact results and different macroscopic regimes. Phys. Rev. E 58 (1998) 1911.