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Modeling snow crystal growth. I: Rigorous results for Packard’s digital snowflakes. (English) Zbl 1122.37057

Summary: Digital snowflakes are solidifying cellular automata on the triangular lattice with the property that a site having exactly one occupied neighbor always becomes occupied at the next time step. We demonstrate that each such rule fills the lattice with an asymptotic density that is independent of the initial finite set. There are some cases in which this density can be computed exactly, and others in which it can only be approximated. We also characterize when the final occupied set comes within a uniformly bounded distance of every lattice point. Other issues addressed include macroscopic dynamics and exact solvability.

MSC:

37B15 Dynamical aspects of cellular automata
68Q80 Cellular automata (computational aspects)
82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
62E20 Asymptotic distribution theory in statistics
11B05 Density, gaps, topology
28A80 Fractals