"Percolation dimension" is introduced in this note. It characterizes certain fractals and its definition is based on the Hausdorff dimension. It is shown that percolation dimension and "boundary dimension" are in a sense independent from the Hausdorff dimension and, therefore, provide an additional tool for classification of ...

We show that the Hausdorff dimension of every level set of iterated Brownian motion is equal to 3/4.

Suppose that X is a two-dimensional Brownian motion. The trace X[0, 1] contains a self-avoiding continuous path whose Hausdorff dimension is equal to 2.