Probability Theory and Related Fields

, Volume 108, Issue 2, pp 153–170

Euclidean models of first-passage percolation

  • C. Douglas Howard
  • Charles M. Newman

DOI: 10.1007/s004400050105

Cite this article as:
Howard, C. & Newman, C. Probab Theory Relat Fields (1997) 108: 153. doi:10.1007/s004400050105


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).

Key words and phrases: First-passage percolationPoisson processVoronoi tesselationshape theoremgeodesic
Mathematics Subject Classification (1991): Primary 60K3560G55; secondary 82D30.

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • C. Douglas Howard
    • 1
  • Charles M. Newman
    • 2
  1. 1.Polytechnic University, 6 Metrotech Center, Brooklyn, NY 11201, USA (e-mail:
  2. 2.Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA (e-mail: