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. © 2012 The Author(s).
|Number of pages||19|
|Journal||Journal of Theoretical Probability|
|Early online date||23 Mar 2012|
|Publication status||Published - Sep 2013|