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). https://doi.org/10.1007/s004400050105
- Key words and phrases: First-passage percolation
- Poisson process
- Voronoi tesselation
- shape theorem
- Mathematics Subject Classification (1991): Primary 60K35
- 60G55; secondary 82D30.