Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks. (English) Zbl 1108.60083

Summary: Fluctuations from a hydrodynamic limit of a one-dimensional asymmetric system come at two levels. On the central limit scale \(n^{1/2}\) one sees initial fluctuations transported along characteristics and no dynamical noise. The second order of fluctuations comes from the particle current across the characteristic. For a system made up of independent random walks we show that the second-order fluctuations appear at scale \(n^{1/4}\) and converge to a certain self-similar Gaussian process. If the system is in equilibrium, this limiting process specializes to fractional Brownian motion with Hurst parameter \(1/4\). This contrasts with the asymmetric exclusion and Hammersley’s process whose second-order fluctuations appear at scale \(n^{1/3}\), as has been discovered through related combinatorial growth models.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F17 Functional limit theorems; invariance principles
82C22 Interacting particle systems in time-dependent statistical mechanics
Full Text: DOI arXiv


[1] Aldous, D. and Diaconis, P. (1995). Hammersley’s interacting particle process and longest increasing subsequences. Probab. Theory Related Fields 103 199–213. · Zbl 0836.60107
[2] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178. JSTOR: · Zbl 0932.05001
[3] Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201
[4] De Masi, A. and Presutti, E. (1991). Mathematical Methods for Hydrodynamic Limits. Springer, Berlin. · Zbl 0754.60122
[5] Durrett, R. (1996). Probability : Theory and Examples , 2nd ed. Duxbury Press, Belmont, CA. · Zbl 1202.60001
[6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes : Characterization and Convergence . Wiley, New York. · Zbl 0592.60049
[7] Ferrari, P. A. and Fontes, L. R. G. (1994). Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22 820–832. JSTOR: · Zbl 0806.60099
[8] Ferrari, P. A. and Fontes, L. R. G. (1994). Shock fluctuations in the asymmetric simple exclusion process. Probab. Theory Related Fields 99 305–319. · Zbl 0801.60094
[9] Hammersley, J. M. (1972). A few seedlings of research. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 1 345–394. Univ. California Press, Berkeley. · Zbl 0236.00018
[10] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476. · Zbl 0969.15008
[11] Kipnis, C. and Landim, C. (1999). Scaling Limits of Interacting Particle Systems . Springer, Berlin. · Zbl 0927.60002
[12] Rezakhanlou, F. (2001). Continuum limit for some growth models. II. Ann. Probab. 29 1329–1372. · Zbl 1081.82016
[13] Rezakhanlou, F. (2002). A central limit theorem for the asymmetric simple exclusion process. Ann. Inst. H. Poincaré Probab. Statist. 38 437–464. · Zbl 1001.60031
[14] Rezakhanlou, F. (2002). Continuum limit for some growth models. Stochastic Process. Appl. 101 1–41. · Zbl 1075.82011
[15] Seppäläinen, T. (1996). A microscopic model for the Burgers equation and longest increasing subsequences. Electron. J. Probab. 1 (electronic). · Zbl 0891.60093
[16] Seppäläinen, T. (1999). Existence of hydrodynamics for the totally asymmetric simple \(K\)-exclusion process. Ann. Probab. 27 361–415. · Zbl 0947.60088
[17] Seppäläinen, T. (2000). Strong law of large numbers for the interface in ballistic deposition. Ann. Inst. H. Poincaré Probab. Statist. 36 691–736. · Zbl 0972.60097
[18] Seppäläinen, T. (2002). Diffusive fluctuations for one-dimensional totally asymmetric interacting random dynamics. Comm. Math. Phys. 229 141–182. · Zbl 1043.82028
[19] Seppäläinen, T. and Krug, J. (1999). Hydrodynamics and platoon formation for a totally asymmetric exclusion model with particlewise disorder. J. Statist. Phys. 95 525–567. · Zbl 0964.82041
[20] Spohn, H. (1991). Large Scale Dynamics of Interacting Particles . Springer, Berlin. · Zbl 0742.76002
[21] Varadhan, S. R. S. (2000). Lectures on hydrodynamic scaling. In Hydrodynamic Limits and Related Topics . Fields Inst. Commun. 27 3–40. Amer. Math. Soc., Providence, RI. · Zbl 1060.82514
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.