Probability Theory and Related Fields

, Volume 108, Issue 2, pp 153-170

First online:

Euclidean models of first-passage percolation

  • C. Douglas HowardAffiliated withPolytechnic University, 6 Metrotech Center, Brooklyn, NY 11201, USA (e-mail: howard@math.poly.edu)
  • , Charles M. NewmanAffiliated withCourant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA (e-mail: newman@cims.nyu.edu)

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Summary.

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).

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