Large-cell Monte Carlo renormalization of irreversible growth processes

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Abstract
We apply Monte Carlo sampling to a recently formulated direct-cell renormalization method for irreversible, disorderly growth processes. We are able to carry out large-cell Monte Carlo renormalization for various nonequilibrium problems based on the formulation dealing with relative probabilities. Specifically, we demonstrate the method by application to the ‘‘true’’ self-avoiding walk and the Eden model of growing animals for d=2, 3, and 4 and to the invasion percolation problem for d=2 and 3. The results are asymptotically in agreement with expectations; however, unexpected complications arise, suggesting the possibility of crossovers, and in any case, demonstrating the danger of using small cells alone, because of the very slow convergence as the cell size b is extrapolated to ∞. We comment on the difficulty of applying the present method to the diffusion-limited-aggregation model.