PC Seminar: Dimension theory of the connected components in Mandelbrot percolation

Abstract: It was shown by Broman, Camia, Joosten and Meester (2013) that there is a strictly positive gap between the values of the Hausdorff and box dimension of the complement of the totally disconnected part of supercritical Mandelbrot percolation. It is also known that its Assouad dimension is the largest possible. For such sets exhibiting dimension gaps there is a meaningful way to interpolate between these notions of dimension using intermediate dimensions and the Assouad spectrum. After a brief overview of dimension theory and Mandelbrot percolation, I will present both dimension spectra of this set based on joint work with Sascha Troscheit.

This is a seminar in our seminar series on Probability and Combinatorics (PC).