Variational principles for Hausdorff and packing dimensions of fractal percolation on self-affine sponges

We establish variational principles for the Hausdorff and packing dimensions of a class of statistically self-affine sponges, including in particular fractal percolation sets obtained from Barański and Gatzouras-Lalley carpets and sponges. Our first step is to compute the Hausdorff and packing dimensions of non-degenerate inhomogeneous Mandelbrot measures supported on the associated random limit sets. This is not a straightforward combination of the existing approaches for the deterministic inhomogeneous Bernoulli measures and the Mandelbrot measures on random Sierpiński sponges; it reveals new structural features. The variational principles rely on a specific subclass of inhomogeneous Mandelbrot measures, which are connected to localized digit frequencies in the underlying coding space. This connection makes it possible to construct effective coverings of the random limit set, leading to sharp upper bounds for its Hausdorff and packing dimensions.