The Annals of Probability

A Hsu–Robbins–Erdős strong law in first-passage percolation

Daniel Ahlberg

Full-text: Open access


Large deviations in the context of first-passage percolation was first studied in the early 1980s by Grimmett and Kesten, and has since been revisited in a variety of studies. However, none of these studies provides a precise relation between the existence of moments of polynomial order and the decay of probability tails. Such a relation is derived in this paper, and is used to strengthen the conclusion of the shape theorem. In contrast to its one-dimensional counterpart—the Hsu–Robbins–Erdős strong law—this strengthening is obtained without imposing a higher-order moment condition.

Article information

Ann. Probab., Volume 43, Number 4 (2015), 1992-2025.

Received: June 2013
Revised: January 2014
First available in Project Euclid: 3 June 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F10: Large deviations 60F15: Strong theorems

First-passage percolation shape theorem large deviations


Ahlberg, Daniel. A Hsu–Robbins–Erdős strong law in first-passage percolation. Ann. Probab. 43 (2015), no. 4, 1992--2025. doi:10.1214/14-AOP926.

Export citation


  • [1] Ahlberg, D. (2015). Asymptotics of first-passage percolation on one-dimensional graphs. Adv. in Appl. Probab. 47 182–209.
  • [2] Ahlberg, D. (2015). Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice. J. Theoret. Probab. 28 198–222.
  • [3] Ahlberg, D. (2008). Asymptotics of first-passage percolation on 1-dimensional graphs. Licentiate thesis, Univ. Gothenburg.
  • [4] Antal, P. and Pisztora, A. (1996). On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24 1036–1048.
  • [5] Benjamini, I., Kalai, G. and Schramm, O. (2003). First passage percolation has sublinear distance variance. Ann. Probab. 31 1970–1978.
  • [6] Chayes, J. T. and Chayes, L. (1986). Percolation and random media. In Phénomènes Critiques, Systèmes Aléatoires, Théories de Jauge, Part I, II (Les Houches, 1984) 1001–1142. North-Holland, Amsterdam.
  • [7] Chow, Y. and Zhang, Y. (2003). Large deviations in first-passage percolation. Ann. Appl. Probab. 13 1601–1614.
  • [8] Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9 583–603.
  • [9] Cranston, M., Gauthier, D. and Mountford, T. S. (2009). On large deviation regimes for random media models. Ann. Appl. Probab. 19 826–862.
  • [10] Erdös, P. (1949). On a theorem of Hsu and Robbins. Ann. Math. Stat. 20 286–291.
  • [11] Erdös, P. (1950). Remark on my paper “On a theorem of Hsu and Robbins.” Ann. Math. Stat. 21 138.
  • [12] Garet, O. and Marchand, R. (2007). Large deviations for the chemical distance in supercritical Bernoulli percolation. Ann. Probab. 35 833–866.
  • [13] Grimmett, G. and Kesten, H. (1984). First-passage percolation, network flows and electrical resistances. Z. Wahrsch. Verw. Gebiete 66 335–366.
  • [14] Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif. 61–110. Springer, New York.
  • [15] Hsu, P. L. and Robbins, H. (1947). Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33 25–31.
  • [16] Kesten, H. (1986). Aspects of first passage percolation. In École d’Été de Probabilités de Saint-Flour, XIV—1984. Lecture Notes in Math. 1180 125–264. Springer, Berlin.
  • [17] Kesten, H. (1993). On the speed of convergence in first-passage percolation. Ann. Appl. Probab. 3 296–338.
  • [18] Kingman, J. F. C. (1968). The ergodic theory of subadditive stochastic processes. J. Roy. Statist. Soc. Ser. B 30 499–510.
  • [19] Richardson, D. (1973). Random growth in a tessellation. Proc. Cambridge Philos. Soc. 74 515–528.
  • [20] Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Inst. Hautes Études Sci. Publ. Math. 81 73–205.
  • [21] Zhang, Y. (2010). On the concentration and the convergence rate with a moment condition in first passage percolation. Stochastic Process. Appl. 120 1317–1341.