Fractal percolation on statistically self-affine carpets

Abstract:

We consider a random self-affine carpet $F$ based on an $n\times m$ subdivision of rectangles and a probability $0<p<1$. Starting by dividing $[0,1]^2$ into an $n\times m$ grid of rectangles and selecting these independently with probability $p$, we then divide the selected rectangles into $n\times m$ subrectangles which are again selected with probability $p$; we continue in this way to obtain a statistically self-affine set $F$. We are particularly interested in topological properties of $F$. We show that the critical value of $p$ above which there is a positive probability that $F$ connects the left and right edges of $[0,1]^2$ is the same as the critical value for $F$ to connect the top and bottom edges of $[0,1]^2$. Once this is established we derive various topological properties of $F$ analogous to those known for self-similar carpets.