Hierarchy in Gilbert's tessellation: Multifractal behavior

We investigate the fragment-size distribution produced by a hierarchical generalization of Gilbert's two-dimensional random tessellation. The study is motivated by experimental data on fragmentation processes, obtained by Kooij and coworkers [S. Kooij et al., Nat. Commun. 12, 2521 (2021)2041-172310.1038/s41467-021-22595-1], where the reasons for which experimental data can exhibit a scale-free behavior resulting in power distributions (for unstressed materials) or exponential distribution with a characteristic length scale (for stressed material) are analyzed in great detail. We study two tessellation models, Dichotomous Gilbert (DG) and Hierarchical Gilbert (HG), which implement different hierarchical splitting rules. Both models generate fragment-area distributions well described by a truncated power-law function, namely a power multiplied by a stretched exponential function or generalized Gamma function. The Gilbert's classic tessellation model gives rise, on the contrary, to a simple exponential distribution. To further characterize the underling complexity, we perform a multifractal analysis of the fragment areas. The results demonstrate that Gilbert's hierarchical fragmentation reveals a multifractal scaling.