Discrete Laplacian Growth: Linear Stability vs Fractal Formation

We introduce stochastic Discrete Laplacian Growth and consider its deterministic continuous version. These are reminiscent respectively to well-known Diffusion Limited Aggregation and Hele-Shaw free boundary problem for the interface propagation. We study correlation between stability of deterministic free-boundary problem and macroscopic fractal growth in the corresponding discrete problem. It turns out that fractal growth in the discrete problem is not influenced by stability of its deterministic version. Using this fact one can easily provide a qualitative analytic description of the Discrete Laplacian Growth.