Probability Theory and Related Fields

, Volume 77, Issue 3, pp 307–324 | Cite as

Connectivity properties of Mandelbrot's percolation process

  • J. T. Chayes
  • L. Chayes
  • R. Durrett


In 1974, Mandelbrot introduced a process in [0, 1]2 which he called “canonical curdling” and later used in this book(s) on fractals to generate self-similar random sets with Hausdorff dimension D∈(0,2). In this paper we will study the connectivity or “percolation” properties of these sets, proving all of the claims he made in Sect. 23 of the “Fractal Geometry of Nature” and a new one that he did not anticipate: There is a probability pc∈(0,1) so that if p<pc then the set is “duslike” i.e., the largest connected component is a point, whereas if ppc (notice the =) opposing sides are connected with positive probability and furthermore if we tile the plane with independent copies of the system then there is with probability one a unique unbounded connected component which intersects a positive fraction of the tiles. More succinctly put the system has a first order phase transition.


Phase Transition Stochastic Process Probability Theory Statistical Theory Hausdorff Dimension 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aizenman, M., Chayes, J.T., Chayes, L., Fröhlich, J., Russo, L.: On a sharp transition from area to perimeter law in a system of random surfaces. Commun. Math. Phys. 92 19–69 (1983)Google Scholar
  2. Billingsley, P.: Ergodic theory and information. New York: Wiley 1965Google Scholar
  3. Harris, T.E.: A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc. 56, 13–20 (1960)Google Scholar
  4. Kahane, P., Peyrière, J.: Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131–145 (1976)Google Scholar
  5. Kesten, H.: The critical probability for bond percolation on the square lattice equals 1/2. Comm. Math. Phys. 74 41–59 (1980)Google Scholar
  6. Kesten, H.: Percolation theory for mathematicians, Boston:Birkhäuser 1982Google Scholar
  7. Mandelbrot, B.: Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. J. Fluid. Mech. 62, 331–358 (1974)Google Scholar
  8. Mandelbrot, B.: The fractal geometry of nature. New York: Freeman 1983Google Scholar
  9. Mauldin, R.D., Williams, S.C.: Random recursive constructions: asymptotic geometric and topological properties. Trans. Am. Math. Soc. 295, 325–346 (1986)Google Scholar
  10. Russo, L.: A note on percolation. Z. Wahrscheinlichkeitstheor. Verw. Geb. 43,39–48 (1978)Google Scholar
  11. Russo, L.: On the critical percolation probabilities. Z. Wahrscheinlichkeitstheor. Verw. Geb. 56, 229–237 (1987)Google Scholar
  12. Seymour, P.D., Welsh, D.J.A: Percolation probabilities on the square lattice. Ann. Discrete Math. 3, 227–245 (1978)Google Scholar
  13. Smythe, R.T., Wierman, J.C:First passage percolation on the square lattice. Lect. Notes Math., vol 671, Berlin Heidelberg New York: Springer 1978Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • J. T. Chayes
    • 1
  • L. Chayes
    • 1
  • R. Durrett
    • 2
  1. 1.Laboratory of Atomic and Solid State PhysicsCornell University, IthacaUSA
  2. 2.Department of MathematicsCornell University IthacaUSA

Personalised recommendations