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We study Mandelbrot's percolation process in dimension $d \geq 2$. Theprocess generates random fractal sets by an iterative procedure which starts bydividing the unit cube $[0,1]^d$ in $N^d$ subcubes, and independently retainingor discarding each subcube with probability $p$ or $1-p$ respectively. Thisstep is then repeated within the retained subcubes at all scales. As $p$ isvaried, there is a percolation phase transition in terms of paths for all $d\geq 2$, and in terms of $(d-1)$-dimensional "sheets" for all $d \geq 3$.  For any $d \geq 2$, we consider the random fractal set produced at thepath-percolation critical value $p_c(N,d)$, and show that the probability thatit contains a path connecting two opposite faces of the cube $[0,1]^d$ tends toone as $N \to \infty$. As an immediate consequence, we obtain that the aboveprobability has a discontinuity, as a function of $p$, at $p_c(N,d)$ for all$N$ sufficiently large. This had previously been proved only for $d=2$ (for any$N \geq 2$). For $d \geq 3$, we prove analogous results for sheet-percolation.  In dimension two, Chayes and Chayes proved that $p_c(N,2)$ converges, as $N\to \infty$, to the critical density $p_c$ of site percolation on the squarelattice. Assuming the existence of the correlation length exponent $\nu$ forsite percolation on the square lattice, we establish the speed of convergenceup to a logarithmic factor. In particular, our results imply that$p_c(N,2)-p_c=(\frac{1}{N})^{1/\nu+o(1)}$ as $N \to \infty$, showing aninteresting relation with near-critical percolation.

Large-N Limit of Crossing Probabilities, Discontinuity, and Asymptotic Behavior of Threshold Values in Mandelbrot's Fractal Percolation Process