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Euclidean models of first-passage percolation


We introduce a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝd. Compared to standard FPP on ℤd, these models have some technical complications but also have the advantage of statistical isotropy. We prove two almost sure results: a shape theorem (where isotropy implies an exact Euclidean ball for the asymptotic shape) and nonexistence of certain doubly infinite geodesics (where isotropy yields a stronger result than in standard FPP).

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Received: 21 May 1996 / In revised form: 19 November 1996

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Howard, C., Newman, C. Euclidean models of first-passage percolation. Probab Theory Relat Fields 108, 153–170 (1997).

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  • Key words and phrases: First-passage percolation
  • Poisson process
  • Voronoi tesselation
  • shape theorem
  • geodesic
  • Mathematics Subject Classification (1991): Primary 60K35
  • 60G55; secondary 82D30.