Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Microscopic concavity and fluctuation bounds in a class of deposition processes

Márton Balázs, Júlia Komjáthy, and Timo Seppäläinen

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Abstract

We prove fluctuation bounds for the particle current in totally asymmetric zero range processes in one dimension with nondecreasing, concave jump rates whose slope decays exponentially. Fluctuations in the characteristic directions have order of magnitude t1/3. This is in agreement with the expectation that these systems lie in the same KPZ universality class as the asymmetric simple exclusion process. The result is via a robust argument formulated for a broad class of deposition-type processes. Besides this class of zero range processes, hypotheses of this argument have also been verified in the authors’ earlier papers for the asymmetric simple exclusion and the constant rate zero range processes, and are currently under development for a bricklayers process with exponentially increasing jump rates.

Résumé

Nous démontrons des bornes sur les fluctuations du courant de particules pour des processus de zero-range unidimensionnels totalement asymétriques avec des taux de sauts concaves dont la pente décroît exponentiellement. Les fluctuations dans la direction des caractéristiques sont de l’ordre t1/3 en accord avec les prédictions de la classe d’universalité de KPZ. Notre résultat est obtenu par un raisonnement robuste qui est formulé pour une classe importante de processus de déposition. Au-delà du processus de zero-range, les hypothèses de notre argument ont aussi été vérifiées dans des articles antérieurs pour le processus d’exclusion simple asymétrique et le processus de zero-range avec taux constants. Ces hypothèses sont en cours de développement pour un processus de déposition avec des taux de sauts dont la croissance est exponentielle.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 1 (2012), 151-187.

Dates
First available in Project Euclid: 23 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1327328018

Digital Object Identifier
doi:10.1214/11-AIHP415

Mathematical Reviews number (MathSciNet)
MR2919202

Zentralblatt MATH identifier
1247.82039

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

Keywords
Interacting particle systems Universal fluctuation bounds t1/3-scaling Second class particle Convexity Asymmetric simple exclusion Zero range process

Citation

Balázs, Márton; Komjáthy, Júlia; Seppäläinen, Timo. Microscopic concavity and fluctuation bounds in a class of deposition processes. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 1, 151--187. doi:10.1214/11-AIHP415. https://projecteuclid.org/euclid.aihp/1327328018


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References

  • [1] E. D. Andjel. Invariant measures for the zero range processes. Ann. Probab. 10 (1982) 525–547.
  • [2] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34 (2006) 1339–1369.
  • [3] J. Baik, P. Deift and K. Johansson. On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999) 1119–1178.
  • [4] J. Baik and E. M. Rains. Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100 (2000) 523–541.
  • [5] M. Balázs. Growth fluctuations in a class of deposition models. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 639–685.
  • [6] M. Balázs, E. Cator and T. Seppäläinen. Cube root fluctuations for the corner growth model associated to the exclusion process. Electron. J. Probab. 11 (2006) 1094–1132 (electronic).
  • [7] M. Balázs and J. Komjáthy. Order of current variance and diffusivity in the rate one totally asymmetric zero range process. J. Stat. Phys. 133 (2008) 59–78.
  • [8] M. Balázs, F. Rassoul-Agha and T. Seppäläinen. The random average process and random walk in a space–time random environment in one dimension. Comm. Math. Phys. 266 (2006) 499–545.
  • [9] 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 (2007) 1201–1249.
  • [10] M. Balázs and T. Seppäläinen. A convexity property of expectations under exponential weights. Available at http://arxiv.org/abs/0707.4273, 2007.
  • [11] M. Balázs and T. Seppäläinen. Exact connections between current fluctuations and the second class particle in a class of deposition models. J. Stat. Phys. 127 (2007) 431–455.
  • [12] M. Balázs and T. Seppäläinen. Fluctuation bounds for the asymmetric simple exclusion process. ALEA Lat. Am. J. Probab. Math. Stat. VI (2009) 1–24.
  • [13] M. Balázs and T. Seppäläinen. Order of current variance and diffusivity in the asymmetric simple exclusion process. Ann. of Math. 171 (2010) 1237–1265.
  • [14] A. Borodin, P. L. Ferrari, M. Prähofer and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007) 1055–1080.
  • [15] E. Cator and P. Groeneboom. Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 (2006) 1273–1295.
  • [16] C. Cocozza-Thivent. Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70 (1985) 509–523.
  • [17] D. Dürr, S. Goldstein and J. Lebowitz. Asymptotics of particle trajectories in infinite one-dimensional systems with collisions. Comm. Pure Appl. Math. 38 (1985) 573–597.
  • [18] P. A. Ferrari and L. R. G. Fontes. Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22 (1994) 820–832.
  • [19] P. A. Ferrari and L. R. G. Fontes. Fluctuations of a surface submitted to a random average process. Electron. J. Probab. 3 (1998) pp. 34 (electronic).
  • [20] P. L. Ferrari and H. Spohn. Scaling limit for the space–time covariance of the stationary totally asymmetric simple exclusion process. Comm. Math. Phys. 265 (2006) 1–44.
  • [21] J. Gravner, C. A. Tracy and H. Widom. Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Stat. Phys. 102 (2001) 1085–1132.
  • [22] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437–476.
  • [23] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003) 277–329.
  • [24] S. Karlin. Total Positivity. Vol. I. Stanford University Press, Stanford, CA, 1968.
  • [25] R. Kumar. Space–time current process for independent random walks in one dimension. ALEA Lat. Am. J. Probab. Math. Stat. IV (2008) 307–336.
  • [26] T. M. Liggett. An infinite particle system with zero range interactions. Ann. Probab. 1 (1973) 240–253.
  • [27] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer-Verlag, New York, 1985.
  • [28] M. Prähofer and H. Spohn. Scale invariance of the PNG droplet and the Airy process. J. Stat. Phys. 108 (2002) 1071–1106. Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65th birthdays.
  • [29] J. Quastel and B. Valkó. t1/3 Superdiffusivity of finite-range asymmetric exclusion processes on ℤ. Comm. Math. Phys. 273 (2007) 379–394.
  • [30] J. Quastel and B. Valkó. A note on the diffusivity of finite-range asymmetric exclusion processes on ℤ. In In and Out Equilibrium 2 543–550. V. Sidoravicius and M. E. Vares (Eds). Progress in Probability 60. Birkhäuser, Basel, 2008.
  • [31] F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on Zd. Comm. Math. Phys. 140 (1991) 417–448.
  • [32] T. Seppäläinen. Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks. Ann. Probab. 33 (2005) 759–797.
  • [33] F. Spitzer. Interaction of Markov processes. Advances in Math. 5 (1970) 246–290.
  • [34] C. A. Tracy and H. Widom. Total current fluctuations in the asymmetric simple exclusion process. J. Math. Phys. 50 095204, 2009.