Skip to main content

Convergence Towards an Asymptotic Shape in First-Passage Percolation on Cone-Like Subgraphs of the Integer Lattice

Abstract

In first-passage percolation on the integer lattice, the shape theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the \({\mathbb {Z}}^d\) lattice, where \(d\ge 2\). In particular, we identify the asymptotic shapes associated with these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for \(L^p\)- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.

This is a preview of subscription content, access via your institution.

References

  1. Ahlberg, D., Damron, M., Sidoravicius, V.: Inhomogeneous first-passage percolation. In preparation (2013)

  2. Ahlberg, D.: A Hsu-Robbins-Erdős strong law in first-passage percolation. Available as arXiv: 1305.6260 (2013)

    Google Scholar 

  3. Ahlberg, D.: Asymptotics of first-passage percolation on 1-dimensional graphs. Available as arXiv: 1107.2276 (2011)

    Google Scholar 

  4. Auffinger, A., Damron, M., Hanson, J.: Limiting geodesics for first-passage percolation on subsets of \(\mathbb{Z}^2\). Available as arXiv: 1302.5413 (2013)

  5. Benjamini, I., Kalai, G., Schramm, O.: Noise sensitivity of boolean functions and applications to percolation. Inst. Hautes Études Sci. Publ. Math. 90, 5–43 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Benjamini, I., Häggström, O., Peres, Y., Steif, J.E.: Which properties of a random sequence are dynamically sensitive? Ann. Probab. 31, 1–34 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cox, J.T.: The time constant of first-passage percolation on the square lattice. Adv. Appl. Probab. 12, 864–879 (1980)

    Article  MATH  Google Scholar 

  8. Cox, J.T., Durrett, R.: Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Probab. 9, 583–603 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cox, J.T., Kesten, H.: On the continuity of the time constant of first-passage percolation. J. Appl. Probab. 18, 809–819 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  10. Garban, C., Pete, G., Schramm, O.: The Fourier spectrum of critical percolation. Acta Math. 205, 19–104 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Grimmett, G.: Bond percolation on subsets of the square lattice, the transition between one-dimensional and two-dimensional behaviour. J. Phys. A 16, 599–604 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  12. Häggström, O., Peres, Y., Steif, J.E.: Dynamical percolation. Ann. Inst. Henri Poincaré Probab. Stat. 33, 497–528 (1997)

    Article  MATH  Google Scholar 

  13. Hammersley, J.M., Welsh, D.J.A.: First-passage percolation, subadditive processes, stochastic networks and generalized renewal theory. In: Neyman, J., Le Cam, L. (eds.) Bernoulli, Bayes, Laplace Anniversary Volume, Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Calif., pp. 61–110. Springer, New York (1965)

  14. Hsu, P.L., Robbins, H.: Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33, 25–31 (1947)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kesten, H.: Aspects of first-passage percolation. In: École d’Été de Probabilités de Saint Flour XIV - 1984, volume 1180 of Lecture Notes in Math., pp. 125–264. Springer, Berlin (1986)

  16. Kingman, J.F.C.: The ergodic theory of subadditive stochastic processes. J. R. Stat. Soc. Ser. B 30, 499–510 (1968)

    MATH  MathSciNet  Google Scholar 

  17. Richardson, D.: Random growth in a tesselation. Proc. Camb. Philos. Soc. 74, 515–528 (1973)

    Article  MATH  Google Scholar 

  18. Schramm, O., Steif, J.E.: Quantitative noise sensitivity and exceptional times for percolation. Ann. Math. 171(2), 619–672 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author would like to thank Olle Häggström for suggesting the dynamical version of first-passage percolation, as well as Robert Morris and Graham Smith for discussing some geometrical issues.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Daniel Ahlberg.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ahlberg, D. Convergence Towards an Asymptotic Shape in First-Passage Percolation on Cone-Like Subgraphs of the Integer Lattice. J Theor Probab 28, 198–222 (2015). https://doi.org/10.1007/s10959-013-0521-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-013-0521-0

Keywords

  • First-passage percolation
  • Shape theorem
  • Large deviations
  • Dynamical stability

Mathematics Subject Classification (2010)

  • Primary 60K35
  • Secondary 82C43
  • 60J25