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

Daniel Ahlberg ()
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Daniel Ahlberg: University of Gothenburg

Journal of Theoretical Probability, 2015, vol. 28, issue 1, 198-222

Abstract: 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 $${\mathbb {Z}}^d$$ Z d lattice, where $$d\ge 2$$ d ≥ 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 $$L^p$$ 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.

Keywords: First-passage percolation; Shape theorem; Large deviations; Dynamical stability; Primary 60K35; Secondary 82C43; 60J25 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s10959-013-0521-0

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