In joint work with Sascha Troscheit (Uppsala), we study supercritical
fractal percolation with inhomogeneous retention probabilities on the
\(M \times N\) anisotropic grid (\(M < N\)). Conditioned on survival of the
limit set, we determine its Assouad dimension and its full Assouad
spectrum, including a detailed description in the critical windows.
Our results show two interesting phenomena.
1. The almost sure constant value of the Assouad dimension exhibits a
nuanced dichotomy depending on whether all its columns are subcritical
or not. In particular, it is not always full even for homogeneous
retention probabilities.
2. In the presence of at least one supercritical column, the Assouad
spectrum has two non-trivial dimension windows, which is the first
non-pathological example showcasing this behaviour.