, Volume 108, Issue 2, pp 153-170

Euclidean models of first-passage percolation

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We introduce a new family of first-passage percolation (FPP) models in the context of Poisson-Voronoi tesselations of ℝ d . Compared to standard FPP on ℤ d , 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).

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