Probability Theory and Related Fields

, Volume 90, Issue 3, pp 291–300 | Cite as

Phase transitions in Mandelbrot's percolation process in three dimensions

  • J. T. Chayes
  • L. Chayes
  • E. Grannan
  • G. Swindle


We study the phase structure and transitions in three-dimensional Mandelbrot percolation—a process which generates random fractal sets. We establish the existence of three distinct phase transitions, and we show that two of these transitions, corresponding to percolation across the initial set by paths and sheets, are discontinuous.


Phase Transition Stochastic Process Probability Theory Phase Structure Mathematical Biology 
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  1. [AG] Aizenman, M., Grimmett, G.R.: Strict inequalities for critical points in percolation and ferromagnetic models. (preprint, 1991)Google Scholar
  2. [AN] Athreya, K.B., Ney, P.: Branching processes. Berlin Heidelberg New York. Springer 1972Google Scholar
  3. [CC] Chayes, J.T., Chayes, L.: The largeN limit of the threshold values in Mandelbrot's percolation process.J. Phys. A. 22, L501-L506 (1989)Google Scholar
  4. [CCD] Chayes, J.T., Chayes, L., Durrett, R.: Connectivity properties of Mandelbrot's percolation process. Probab. Th. Rel. Fields77, 307–324 (1988)Google Scholar
  5. [DG] Dekking, F.M., Grimmett, G.R.: Superbranching processes and projections of random cantor sets Probab. Th. Rel. Fields78, 335–355 (1988)Google Scholar
  6. [DM] Dekking, F.M., Meester, R.W.J.: On the structure of Mandelbrot's percolation process and other random cantor sets. (preprint, 1991)Google Scholar
  7. [D] Durrett, R.: Lecture notes on particle systems and percolation. Belmont, CA: Wadsworth 1988Google Scholar
  8. [GS] Grannan, E., Shors, D.: A duality result in graph theory for percolation in three dimensions. (preprint)Google Scholar
  9. [M] Mandelbrot, B.: The fractal geometry of nature. New York: W.H. Freemand 1983Google Scholar

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • J. T. Chayes
    • 1
  • L. Chayes
    • 1
  • E. Grannan
    • 2
  • G. Swindle
    • 3
  1. 1.Department of MathematicsUniversity of CaliforniaLos AngelesUSA
  2. 2.AT & T Bell LaboratoriesMurray HillUSA
  3. 3.Department of Statistics and Applied ProbabilityUniversity of CaliforniaSanta BarbaraUSA

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