A discrete time evolution model for fracture networks

We examine geological crack patterns using the mean field theory of convex mosaics. We assign the pair n over bar *,v over bar *\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left({\overline{n } }<^>{*},{\overline{v } }<^>{*}\right)$$\end{document} of average corner degrees (Domokos et al. in A two-vertex theorem for normal tilings. Aequat Math , 2022) to each crack pattern and we define two local, random evolutionary steps R-0 and R-1, corresponding to secondary fracture and rearrangement of cracks, respectively. Random sequences of these steps result in trajectories on the n over bar *,v over bar *\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left({\overline{n } }<^>{*},{\overline{v } }<^>{*}\right)$$\end{document} plane. We prove the existence of limit points for several types of trajectories. Also, we prove that celldensity rho over bar =v over bar *n over bar *\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\rho }= \frac{{\overline{v } }<^>{*}}{{\overline{n } }<^>{*}}$$\end{document} increases monotonically under any admissible trajectory.