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Microscopic concavity and fluctuation bounds in a class of deposition processes. (English. French summary) Zbl 1247.82039
The paper addresses anomalous current fluctuations of attractive interacting systems in one dimension, with one conserved quantity, specifically the hydrodynamic flux function which determines the \(t\)-scaling. The paper focuses on the \(1/3\) exponent and makes a further step towards establishing the universality of \(t^{1/3}\)-order fluctuations, by rewriting an earlier proof for the asymmetric simple exclusion process (ASEP) and the constant rate totally asymmetric zero range process (TAZRP) in a fairly general way. A crucial feature that underlies scaling proofs connected with these models is named microscopic concavity property. Considered as a hypothesis, this property may be interpreted as an input that relaxes otherwise very rigid earlier proofs of the \(t^{1/3}\) scaling. Three subclasses of processes are identified to respect this property, e.g., ASEP, TAZRP and the totally asymmetric bricklayers process with convex exponential jump rate. The authors expect the validity of the above-mentioned hypothesis for a broader class of totally asymmetric concave zero range processes, because its key part can be directly verified and only a certain tail estimate is missing (to be examined in case of a specific model).

82C22 Interacting particle systems in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
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[1] E. D. Andjel. Invariant measures for the zero range processes. Ann. Probab. 10 (1982) 525-547. · Zbl 0492.60096
[2] C. Bahadoran, H. Guiol, K. Ravishankar and E. Saada. Euler hydrodynamics of one-dimensional attractive particle systems. Ann. Probab. 34 (2006) 1339-1369. · Zbl 1101.60075
[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. · Zbl 0932.05001
[4] J. Baik and E. M. Rains. Limiting distributions for a polynuclear growth model with external sources. J. Stat. Phys. 100 (2000) 523-541. · Zbl 0976.82043
[5] M. Balázs. Growth fluctuations in a class of deposition models. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 639-685. · Zbl 1029.60075
[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). · Zbl 1139.60046
[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. · Zbl 1151.82381
[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. · Zbl 1129.60097
[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. · Zbl 1138.60340
[10] M. Balázs and T. Seppäläinen. A convexity property of expectations under exponential weights. Available at , 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. · Zbl 1147.82348
[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. · Zbl 1160.60333
[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. · Zbl 1200.60083
[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. · Zbl 1136.82028
[15] E. Cator and P. Groeneboom. Second class particles and cube root asymptotics for Hammersley’s process. Ann. Probab. 34 (2006) 1273-1295. · Zbl 1101.60076
[16] C. Cocozza-Thivent. Processus des misanthropes. Z. Wahrsch. Verw. Gebiete 70 (1985) 509-523. · Zbl 0554.60097
[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. · Zbl 0578.60094
[18] P. A. Ferrari and L. R. G. Fontes. Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22 (1994) 820-832. · Zbl 0806.60099
[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). · Zbl 0903.60089
[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. · Zbl 1118.82032
[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. · Zbl 0989.82030
[22] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437-476. · Zbl 0969.15008
[23] K. Johansson. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003) 277-329. · Zbl 1031.60084
[24] S. Karlin. Total Positivity. Vol. I . Stanford University Press, Stanford, CA, 1968. · Zbl 0219.47030
[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. · Zbl 1162.60345
[26] T. M. Liggett. An infinite particle system with zero range interactions. Ann. Probab. 1 (1973) 240-253. · Zbl 0264.60083
[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. · Zbl 1025.82010
[29] J. Quastel and B. Valkó. t 1/3 Superdiffusivity of finite-range asymmetric exclusion processes on \Bbb Z. Comm. Math. Phys. 273 (2007) 379-394. · Zbl 1127.60091
[30] J. Quastel and B. Valkó. A note on the diffusivity of finite-range asymmetric exclusion processes on \Bbb Z. In In and Out Equilibrium 2 543-550. V. Sidoravicius and M. E. Vares (Eds). Progress in Probability 60 . Birkhäuser, Basel, 2008. · Zbl 1173.82341
[31] F. Rezakhanlou. Hydrodynamic limit for attractive particle systems on Z d . Comm. Math. Phys. 140 (1991) 417-448. · Zbl 0738.60098
[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. · Zbl 1108.60083
[33] F. Spitzer. Interaction of Markov processes. Advances in Math. 5 (1970) 246-290. · Zbl 0312.60060
[34] C. A. Tracy and H. Widom. Total current fluctuations in the asymmetric simple exclusion process. J. Math. Phys. 50 095204, 2009. · Zbl 1241.82051
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