Abstract
We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is almost surely the union of non-trivial Hölder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.
Original language | English (US) |
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Pages (from-to) | 836-854 |
Number of pages | 19 |
Journal | Journal of Theoretical Probability |
Volume | 26 |
Issue number | 3 |
DOIs | |
State | Published - Sep 2013 |
Keywords
- Box counting dimension
- Fractal percolation
- Hausdorff dimension
- Hölder continuous curves
- Subsequential weak limits
ASJC Scopus subject areas
- Statistics and Probability
- General Mathematics
- Statistics, Probability and Uncertainty