# Microscopic concavity and fluctuation bounds in a class of deposition processes

Márton Balázs; Júlia Komjáthy; Timo Seppäläinen

Annales de l'I.H.P. Probabilités et statistiques (2012)

- Volume: 48, Issue: 1, page 151-187
- ISSN: 0246-0203

## Access Full Article

top## Abstract

top## How to cite

topBalázs, Márton, Komjáthy, Júlia, and Seppäläinen, Timo. "Microscopic concavity and fluctuation bounds in a class of deposition processes." Annales de l'I.H.P. Probabilités et statistiques 48.1 (2012): 151-187. <http://eudml.org/doc/272024>.

@article{Balázs2012,

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.},

author = {Balázs, Márton, Komjáthy, Júlia, Seppäläinen, Timo},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {interacting particle systems; universal fluctuation bounds; t1/3-scaling; second class particle; convexity; asymmetric simple exclusion; zero range process; scaling; KPZ model; microscopic concavity hypothesis; Tracy-Widom distributions and processes; determinantal representation},

language = {eng},

number = {1},

pages = {151-187},

publisher = {Gauthier-Villars},

title = {Microscopic concavity and fluctuation bounds in a class of deposition processes},

url = {http://eudml.org/doc/272024},

volume = {48},

year = {2012},

}

TY - JOUR

AU - Balázs, Márton

AU - Komjáthy, Júlia

AU - Seppäläinen, Timo

TI - Microscopic concavity and fluctuation bounds in a class of deposition processes

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2012

PB - Gauthier-Villars

VL - 48

IS - 1

SP - 151

EP - 187

AB - 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.

LA - eng

KW - interacting particle systems; universal fluctuation bounds; t1/3-scaling; second class particle; convexity; asymmetric simple exclusion; zero range process; scaling; KPZ model; microscopic concavity hypothesis; Tracy-Widom distributions and processes; determinantal representation

UR - http://eudml.org/doc/272024

ER -

## References

top- [1] E. D. Andjel. Invariant measures for the zero range processes. Ann. Probab.10 (1982) 525–547. Zbl0492.60096MR659526
- [2] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab.34 (2006) 1339–1369. Zbl1101.60075MR2257649
- [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. Zbl0932.05001MR1682248
- [4] J. Baik and E. M. Rains. Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys.100 (2000) 523–541. Zbl0976.82043MR1788477
- [5] M. Balázs. Growth fluctuations in a class of deposition models. Ann. Inst. H. Poincaré Probab. Statist.39 (2003) 639–685. Zbl1029.60075MR1983174
- [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). Zbl1139.60046MR2268539
- [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. Zbl1151.82381MR2438897
- [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. Zbl1129.60097MR2238887
- [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. Zbl1138.60340MR2330972
- [10] M. Balázs and T. Seppäläinen. A convexity property of expectations under exponential weights. Available at http://arxiv.org/abs/0707.4273, 2007. Zbl1147.82348
- [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. Zbl1147.82348MR2314355
- [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. Zbl1160.60333MR2485877
- [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. Zbl1200.60083MR2630064
- [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. Zbl1136.82028MR2363389
- [15] E. Cator and P. Groeneboom. Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab.34 (2006) 1273–1295. Zbl1101.60076MR2257647
- [16] C. Cocozza-Thivent. Processus des misanthropes. Z. Wahrsch. Verw. Gebiete70 (1985) 509–523. Zbl0554.60097MR807334
- [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. Zbl0578.60094MR803248
- [18] P. A. Ferrari and L. R. G. Fontes. Current fluctuations for the asymmetric simple exclusion process. Ann. Probab.22 (1994) 820–832. Zbl0806.60099MR1288133
- [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). Zbl0903.60089MR1624854
- [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. Zbl1118.82032MR2217295
- [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. Zbl0989.82030MR1830441
- [22] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys.209 (2000) 437–476. Zbl0969.15008MR1737991
- [23] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys.242 (2003) 277–329. Zbl1031.60084MR2018275
- [24] S. Karlin. Total Positivity. Vol. I. Stanford University Press, Stanford, CA, 1968. Zbl0219.47030MR230102
- [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. Zbl1162.60345MR2456971
- [26] T. M. Liggett. An infinite particle system with zero range interactions. Ann. Probab.1 (1973) 240–253. Zbl0264.60083MR381039
- [27] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer-Verlag, New York, 1985. Zbl0559.60078MR776231
- [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. Zbl1025.82010MR1933446
- [29] J. Quastel and B. Valkó. t1/3 Superdiffusivity of finite-range asymmetric exclusion processes on ℤ. Comm. Math. Phys. 273 (2007) 379–394. Zbl1127.60091MR2318311
- [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. Zbl1173.82341MR2477398
- [31] F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on Zd. Comm. Math. Phys.140 (1991) 417–448. Zbl0738.60098MR1130693
- [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. Zbl1108.60083MR2123209
- [33] F. Spitzer. Interaction of Markov processes. Advances in Math.5 (1970) 246–290. Zbl0312.60060MR268959
- [34] C. A. Tracy and H. Widom. Total current fluctuations in the asymmetric simple exclusion process. J. Math. Phys. 50 095204, 2009. Zbl1241.82051MR2566884

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.