The Power Laws of Geodesics in Some Random Sets with Dilute Concentration of Inclusions

  • François WillotEmail author
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9082)


A method for computing upper-bounds on the length of geodesics spanning random sets in 2D and 3D is proposed, with emphasis on Boolean models containing a vanishingly small surface or volume fraction of inclusions f ≪ 1. The distance function is zero inside the grains and equal to the Euclidean distance outside of them, and the geodesics are shortest paths connecting two points far from each other. The asymptotic behavior of the upper-bounds is derived in the limit f → 0. The scalings involve powerlaws with fractional exponents ~f 2/3 for Boolean sets of disks or aligned squares and ~f 1/2 for the Boolean set of spheres. These results are extended to models of hyperspheres in arbitrary dimension and, in 2D and 3D, to a more general problem where the distance function is non-zero in the inclusions. Finally, other fractional exponents are derived for the geodesics spanning multiscale Boolean sets, based on inhomogeneous Poisson point processes, in 2D and 3D.


Geodesic Shortest paths Stochastic geometry Boolean models Multiscale random sets 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Li, F., Klette, R.: Euclidean shortest paths – exact or approximate algorithms. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  2. 2.
    Roux, S., Herrmann, H.J.: Disordered-induced nonlinear conductivity. Europhys. Lett. 4(11), 1227–1231 (1987)CrossRefGoogle Scholar
  3. 3.
    Roux, S., Hansen, A., Guyon, É.: Criticality in non-linear transport properties of heteroegeneous materials. J. Physique 48(12), 2125–2130 (1987)CrossRefGoogle Scholar
  4. 4.
    Roux, S., Herrmann, H., Hansen, A., Guyon, É.: Relation entre différents types de comportements non linéaires de réseaux désordonnés. C. R. Acad. Sci. Série II 305(11), 943–948 (1987)Google Scholar
  5. 5.
    Roux, S., François, D.: A simple model for ductile fracture of porous materials. Scripta Metall. Mat. 25(5), 1087–1092 (1991)CrossRefGoogle Scholar
  6. 6.
    Derrida, B., Vannimenus, J.: Interface energy in random systems. Phys. Rev. B 27(7), 4401 (1983)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Willot, F.: The power law of geodesics in 2D random media with dilute concentration of disks. Submitted to Phys. Rev. E.Google Scholar
  8. 8.
    Willot, F.: Contribution à l’étude théorique de la localisation plastique dans les poreux. Diss., École Polytechnique (2007)Google Scholar
  9. 9.
    Matheron, G.: Random sets and integral geometry. Wiley, New-York (1975)zbMATHGoogle Scholar
  10. 10.
    Matheron, G.: Random sets theory and its applications to stereology. J. Microscopy 95(1), 15–23 (1972)CrossRefMathSciNetGoogle Scholar
  11. 11.
    MATLAB and Statistics Toolbox Release 2012b, The MathWorks, Inc., Natick, Massachusetts, United StatesGoogle Scholar
  12. 12.
    Soille, P.: Generalized geodesy vis geodesic time. Pat. Rec. Let. 15(12), 1235–1240 (1994)CrossRefGoogle Scholar
  13. 13.
    Quintanilla, J., Torquato, S., Ziff, R.M.: Efficient measurement of the percolation threshold for fully penetrable discs. J. Phys. A: Math. Gen. 33(42), L399 (2000)Google Scholar
  14. 14.
    Wolfram Research, Inc.: Mathematica, Version 10.0. Champaign, IL (2014)Google Scholar
  15. 15.
    Torquato, S.: Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes. J. of Chem. Phys. 136(5), 054106 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Center for Mathematical Morphology, Mines ParisTechPSL Research UniversityFontainebleauFrance

Personalised recommendations