Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Hammersley’s harness process: Invariant distributions and height fluctuations

Timo Seppäläinen and Yun Zhai

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the invariant distributions of Hammersley’s serial harness process in all dimensions and height fluctuations in one dimension. Subject to mild moment assumptions there is essentially one unique invariant distribution, and all other invariant distributions are obtained by adding harmonic functions of the averaging kernel. We identify one Gaussian case where the invariant distribution is i.i.d. Height fluctuations in one dimension obey the stochastic heat equation with additive noise (Edwards–Wilkinson universality). We prove this for correlated initial data subject to fast enough polynomial decay of strong mixing coefficients, including process-level tightness in the Skorohod space of space–time trajectories.

Résumé

Nous étudions la mesure invariante du processus de harnais de Hammersley en dimension arbitraire et les fluctuations de la hauteur en dimension un. Sous des hypothèses douces sur les moments, il y a essentiellement une mesure invariante unique, et toutes les autres mesures invariantes sont obtenues par l’addition de fonctions harmoniques du noyau. Nous identifions un cas Gaussien ou la mesure invariante est i.i.d. Les fluctuations de la hauteur en dimension un obéissent à l’équation stochastique de chaleur à bruit additif (universalité d’Edwards–Wilkinson). Nous démontrons ce résultat dans le cas de données initiales corrélées dont les coefficients de mélange fort décroissent assez rapidement, y compris l’étroitesse au niveau du processus dans l’espace de Skorohod de trajectoires spatio-temporelles.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 1 (2017), 287-321.

Dates
Received: 12 June 2015
Revised: 27 September 2015
Accepted: 30 September 2015
First available in Project Euclid: 8 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1486544893

Digital Object Identifier
doi:10.1214/15-AIHP717

Mathematical Reviews number (MathSciNet)
MR3606743

Zentralblatt MATH identifier
1361.60096

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60F05: Central limit and other weak theorems

Keywords
Harness Gaussian process Edwards–Wilkinson universality class Random walk Fractional Brownian motion Fluctuations Interface Process tightness Strong mixing coefficients Linear process Harmonic crystal Stochastic heat equation

Citation

Seppäläinen, Timo; Zhai, Yun. Hammersley’s harness process: Invariant distributions and height fluctuations. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 1, 287--321. doi:10.1214/15-AIHP717. https://projecteuclid.org/euclid.aihp/1486544893


Export citation

References

  • [1] R. Arratia. The motion of a tagged particle in the simple symmetric exclusion system on $\mathbf{Z}$. Ann. Probab. 11 (2) (1983) 362–373.
  • [2] 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 (2) (2006) 499–545.
  • [3] P. J. Bickel and M. J. Wichura. Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat. 42 (1971) 1656–1670.
  • [4] A. Borodin and V. Gorin. Lectures on integrable probability. Available at arXiv:1212.3351.
  • [5] R. C. Bradley. Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 (2005) 107–144. Update of, and a supplement to, the 1986 original.
  • [6] P. Caputo and J.-D. Deuschel. Large deviations and variational principle for harmonic crystals. Comm. Math. Phys. 209 (3) (2000) 595–632.
  • [7] I. Corwin. The Kardar–Parisi–Zhang equation and universality class. Random Matrices Theory Appl. 1 (1) (2012) 1130001.
  • [8] A. De Masi and P. A. Ferrari. Flux fluctuations in the one dimensional nearest neighbors symmetric simple exclusion process. J. Stat. Phys. 107 (3–4) (2002) 677–683.
  • [9] D. Dürr, S. Goldstein and J. Lebowitz. Asymptotics of particle trajectories in infinite one-dimensional systems with collisions. Comm. Pure Appl. Math. 38 (5) (1985) 573–597.
  • [10] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010.
  • [11] W. Feller. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York, 1971.
  • [12] P. A. Ferrari and L. R. G. Fontes. Fluctuations of a surface submitted to a random average process. Electron. J. Probab. 3 (1998) 6 (electronic).
  • [13] P. A. Ferrari and B. M. Niederhauser. Harness processes and harmonic crystals. Stochastic Process. Appl. 116 (6) (2006) 939–956.
  • [14] N. G. Gamkrelidze. A measure of “smoothness” of multidimensional distributions of integer-valued random vectors. Theory Probab. Appl. 30 (2) (1985) 427–431.
  • [15] J. M. Hammersley. Harnesses. In Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability (Berkeley, CA, 1965/66), Vol. III: Physical Sciences 89–117. University of California Press, Berkeley, CA, 1967.
  • [16] T. E. Harris. Diffusion with “collisions” between particles. J. Appl. Probab. 2 (1965) 323–338.
  • [17] C. T. Hsiao. Stochastic processes with Gaussian interaction of components. Z. Wahrsch. Verw. Gebiete 59 (1) (1982) 39–53.
  • [18] C. T. Hsiao. Infinite systems with locally additive interaction of components. Chinese J. Math. 13 (2) (1985) 83–95.
  • [19] M. D. Jara and C. Landim. Nonequilibrium central limit theorem for a tagged particle in symmetric simple exclusion. Ann. Inst. Henri Poincaré Probab. Stat. 42 (5) (2006) 567–577.
  • [20] J. Mathew, F. Rassoul-Agha and T. Seppäläinen. Independent particles in a dynamical random environment. Available at arXiv:1110.1889.
  • [21] R. Kumar. Space–time current process for independent random walks in one dimension. ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008) 307–336.
  • [22] G. F. Lawler and V. Limic. Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics 123. Cambridge University Press, Cambridge, 2010.
  • [23] T. M. Liggett. Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer-Verlag, New York, 1985.
  • [24] T. M. Liggett. Interacting particle systems – An introduction. In School and Conference on Probability Theory 1–56. ICTP Lecture Notes XVII. Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004 (electronic).
  • [25] M. Peligrad and S. Sethuraman. On fractional Brownian motion limits in one dimensional nearest-neighbor symmetric simple exclusion. ALEA Lat. Am. J. Probab. Math. Stat. 4 (2008) 245–255.
  • [26] M. Peligrad and S. Utev. Central limit theorem for linear processes. Ann. Probab. 25 (1) (1997) 443–456.
  • [27] J. Peterson and T. Seppäläinen. Current fluctuations of a system of one-dimensional random walks in random environment. Ann. Probab. 38 (6) (2010) 2258–2294.
  • [28] M. A. Pinsky. Introduction to Fourier Analysis and Wavelets. Brooks/Cole Series in Advanced Mathematics. Brooks/Cole, Pacific Grove, CA, 2002.
  • [29] E. Rio. Inequalities and limit theorems for weakly dependent sequences. Lecture notes for 3ème cycle. Available at https://cel.archives-ouvertes.fr/cel-00867106, 2013.
  • [30] H. Rost and M. E. Vares. Hydrodynamics of a one-dimensional nearest neighbor model. In Particle Systems, Random Media and Large Deviations 329–342. Contemp. Math. 41. Amer. Math. Soc., Providence, RI, 1985.
  • [31] T. Seppäläinen. Translation invariant exclusion processes. Lecture notes. Available at http://www.math.wisc.edu/~seppalai/excl-book/etusivu.html.
  • [32] T. Seppäläinen. Second-order fluctuations and current across characteristic for a one-dimensional growth model of independent random walks. Ann. Probab. 33 (2) (2005) 759–797.
  • [33] T. Seppäläinen. Current Fluctuations for Stochastic Particle Systems with Drift in One Spatial Dimension. Ensaios Matemáticos [Mathematical Surveys] 18. Sociedade Brasileira de Matemática, Rio de Janeiro, 2010.
  • [34] F. Spitzer. Principles of Random Walks, 2nd edition. Graduate Texts in Mathematics 34. Springer-Verlag, New York, 1976.
  • [35] A. Toom. Tails in harnesses. J. Stat. Phys. 88 (1–2) (1997) 347–364.
  • [36] J. B. Walsh. An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour, XIV – 1984 265–439. Lecture Notes in Mathematics 1180. Springer, Berlin, 1986.
  • [37] J. Yu. Edwards–Wilkinson fluctuations in the Howitt–Warren flows. Available at arXiv:1412.3911.
  • [38] Y. Zhai. Discrete time harness processes. Ph.D. thesis, University of Wisconsin, 2015. Available at arXiv:1506.02116.