Convergence Towards an Asymptotic Shape in First-Passage Percolation on Cone-Like Subgraphs of the Integer Lattice
Journal of Theoretical Probability(2013)
摘要
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 ℤ^d lattice, where 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 - 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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关键词
First-passage percolation,Shape theorem,Large deviations,Dynamical stability
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