Advances in Applied Probability

Shape theorems for Poisson hail on a bivariate ground

François Baccelli, Héctor A. Chang-Lara, and Sergey Foss

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


We consider an extension of the Poisson hail model where the service speed is either 0 or ∞ at each point of the Euclidean space. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process, and on the geometrical properties of the zero speed set.

Article information

Adv. in Appl. Probab., Volume 48, Number 2 (2016), 525-543.

First available in Project Euclid: 9 June 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60F15: Strong theorems 60G55: Point processes

Point process theory Poisson rain stochastic geometry random closed set time and space growth shape queueing theory max-plus algebra heaps branching process sub-additive ergodic theory


Baccelli, François; Chang-Lara, Héctor A.; Foss, Sergey. Shape theorems for Poisson hail on a bivariate ground. Adv. in Appl. Probab. 48 (2016), no. 2, 525--543.

Export citation


  • Baccelli, F. and Foss, S. (2011). Poisson hail on a hot ground. In New Frontiers in Applied Probability: A Festschrift for Søren Asmussen (J. Appl. Prob. Spec. Vol. 48A), Applied Probability Trust, Sheffield, pp. 343–366.
  • Baccelli, F., Borovkov, A. and Mairesse, J. (2000). Asymptotic results on infinite tandem queueing networks. Prob. Theory Relat. Fields 118, 365–405.
  • Baccelli, F., Cohen, G., Olsder, G. J. and Quadrat, J.-P. (1992). Synchronization and Linearity: An Algebra for Discrete Event Systems. John Wiley, Chichester.
  • Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Prob. 20, 1907–1965.
  • Cox, J. T., Gandolfi, A., Griffin, P. S. and Kesten, H. (1993). Greedy lattice animals. I. Upper bounds. Ann. Appl. Prob. 3, 1151–1169.
  • Foss, S., Konstantopoulos, T. and Mountford, T. (2014). Power law condition for stability of Poisson hail. Preprint. Available at
  • Gandolfi, A. and Kesten, H. (1994). Greedy lattice animals. II. Linear growth. Ann. Appl. Prob. 4, 76–107.
  • Halsey, T. (2000). Diffusion-limited aggregation: a model for pattern formation. Physics Today 53, 36–41.
  • Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Prob. 13, 1279–1285.
  • Seppäläinen, T. (1998). Hydrodynamic scaling, convex duality, and asymptotic shapes of growth models. Markov Process. Relat. Fields 4, 1–26.