Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Hammersley’s harness process: Invariant distributions and height fluctuations

Timo Seppäläinen and Yun Zhai

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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.


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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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