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Keywords:

  • cycle popping;
  • internal diffusion limited aggregation;
  • least action principle;
  • low discrepancy random stack;
  • odometer function;
  • potential kernal;
  • rotor-router model

Abstract

We give an algorithm that computes the final state of certain growth models without computing all intermediate states. Our technique is based on a “least action principle” which characterizes the odometer function of the growth process. Starting from an approximation for the odometer, we successively correct under- and overestimates and provably arrive at the correct final state. Internal diffusion-limited aggregation (IDLA) is one of the models amenable to our technique. The boundary fluctuations in IDLA were recently proved to be at most logarithmic in the size of the growth cluster, but the constant in front of the logarithm is still not known. As an application of our method, we calculate the size of fluctuations over two orders of magnitude beyond previous simulations, and use the results to estimate this constant. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012