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

Journal of Theoretical Probability volume 28, pages 198–222 (2015)Cite this article

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.

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Gothenburg, Gothenburg, Sweden

    Daniel Ahlberg

  2. Department of Mathematical Sciences, Chalmers University of Technology, Gothenburg, Sweden

    Daniel Ahlberg

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  1. Daniel Ahlberg
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Correspondence to Daniel Ahlberg.

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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

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  • 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