Isotropic growth on a grid

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Isotropic growth from a single point on a two-dimensional square grid should generate an increasing sequence of discretized discs. We present a simple probabilistic model for growth on a grid, and discuss a class of parameterizations of the model (called kernels) which was conjectured [S. Thompson and A. Rosenfeld, Discrete stochastic growth models for two-dimensional shapes. In Shape in Picture—Mathematical Descriptions of Shape in Grey level Images, Y. L. O. A. Toet, D. H. Foster and P. Meer (eds), 301–318, Springer-Verlag, Heidelberg (1993)] to produce isotropic growth. We disprove this conjecture, but we claim that these kernels produce growth that can be decomposed into isotropic and nonisotropic other probabilistic growth processes on grids, and describe qualitative and quantitative properties of the models. We also consider a deterministic growth model based on the diffusion equation, and show empirically that discretization of this model leads to a steady state configuration that appears to be polygonal.