Abstract
We relate various concepts of fractal dimension of the limiting set $\mathcal{C}$ in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in $\mathcal{C}$ (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.
Highlights
Introduction and Main ResultsIn this paper we are concerned with a percolation model, first introduced in [12], which is known as Mandelbrot’s fractal percolation process and which can be informally described as follows
The Hausdorff dimension of the limiting set in fractal percolation is a.s. given by the following equation, whose proof can be found in [8] or [11], Proposition 15.4: dimH(C) =
We describe how the existence of subsequential weak limits of the sequence μn follows from this lemma. (This is well known but perhaps not immediately obvious from the literature, our summary for the convenience of the reader.) For
Summary
In this paper we are concerned with a percolation model, first introduced in [12], which is known as Mandelbrot’s fractal percolation process and which can be informally described as follows. Let CR([0, 1]d ) denote the event that C contains a connected component which intersects the left-hand side {0} × [0, 1]d−1 of the unit cube and intersects the right-hand side {1} × [0, 1]d−1. The Hausdorff dimension of the limiting set in fractal percolation is a.s. given by the following equation, whose proof can be found in [8] or [11], Proposition 15.4: dimH(C) =. Their paper deals with scaling limits of systems of random curves, but we will show how their results can be useful in the context of fractal percolation as well. In order for our interface curves to be uniquely defined, we orient them in such a way that they have black (retained) squares on the left and white (discarded) squares on the right, and assume that they turn to the right at corners where two white and two black squares meet in a checkerboard configuration, see Fig. 1. Assume that there exists x ∈ C such that d(x, Mδ i=1
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