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No directed fractal percolation in zero area

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Abstract

We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {p k}; that is, for allk ⩾ 1, at thek th stage every retained square of side lengthN 1− k is partitioned intoN 2 congruent subsquares, and each of these is retained with probabilityp k. independent of all others. We show that if Πk p k =0 (i.e., if the area of the limiting set vanishes a.s.), then a.s. the limiting set contains no directed crossings of the unit square (a directed crossing is a path that crosses the unit square from left to right, and moves only up, down, and to the right).

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References

  1. L. Chayes, On The Absence of Directed Fractal Percolation,J. Phys. A: Math. Gen. 28:L295-L301 (1995).

    Article  MathSciNet  ADS  Google Scholar 

  2. J. T. Chayes, L. Chayes, and R. Durrett, Connectivity Properties of Mandelbrot’s Percolation Process,Probab. Th. Rel. Fields 77:307–324 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  3. B. Duplantier, G. F. Lawler, J.-F. Le Gall, and T. J. Lyons, The Geometry of The Brownian curve,Bull. Sci. Math. 2 e série117:91–106 (1993).

    MATH  Google Scholar 

  4. R. Durrett, Oriented percolation in two dimensions,Ann. Probab. 12:999–1040 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Lyons, R. Pemantle, and Y. Peres, Resistance bounds for first-passage percolation and maximum flow, in preparation (1996).

  6. R. Pemantle, The probability that Brownian motion almost contains a line,Ann. Inst. Henri Poincaré, Probab. et Statist. 33:147–165 (1997).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Y. Peres, Intersection-equivalence of Brownian paths and certain branching processes,Comm. Math. Phys. 177:417–434 (1996).

    Article  MathSciNet  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, 90024, Los Angles, California

    L. Chayes

  2. Department of Mathematics, University of Wisconsin, 53706, Madison, Wisconsin

    Robin Pemantle

  3. Mathematics Institute, The Hebrew University, Jerusalem, Israel

    Yuval Peres

  4. Department of Statistics, University of California, Berkeley, California

    Yuval Peres

Authors
  1. L. Chayes
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  2. Robin Pemantle
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  3. Yuval Peres
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Chayes, L., Pemantle, R. & Peres, Y. No directed fractal percolation in zero area. J Stat Phys 88, 1353–1362 (1997). https://doi.org/10.1007/BF02732437

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  • DOI: https://doi.org/10.1007/BF02732437

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