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

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• 作者：Daniel Ahlberg (1) (2)
  
  1. Department of Mathematical Sciences ; University of Gothenburg ; Gothenburg ; Sweden
  2. Department of Mathematical Sciences ; Chalmers University of Technology ; Gothenburg ; Sweden
  
• 关键词：First ; passage percolation ; Shape theorem ; Large deviations ; Dynamical stability ; Primary 60K35 ; Secondary 82C43 ; 60J25
• 刊名：Journal of Theoretical Probability
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：28
• 期：1
• 页码：198-222
• 全文大小：285 KB
• 参考文献：1. Ahlberg, D., Damron, M., Sidoravicius, V.: Inhomogeneous first-passage percolation. In preparation (2013)
  2. Ahlberg, D.: A Hsu-Robbins-Erd艖s strong law in first-passage percolation. Available as arXiv: 1305.6260 (2013)
  3. Ahlberg, D.: Asymptotics of first-passage percolation on 1-dimensional graphs. Available as arXiv: 1107.2276 (2011)
  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)
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  14. Hsu, PL, Robbins, H (1947) Complete convergence and the law of large numbers. Proc. Natl. Acad. Sci. USA 33: pp. 25-31 CrossRef
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Probability Theory and Stochastic Processes
  Statistics
  
• 出版者：Springer Netherlands
• ISSN：1572-9230

文摘

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.