"Go with the winners"-Simulations

Abstract

We describe a general strategy for sampling configurations from a given (Gibbs-Boltzmann or other) distribution. It is {\it not} based on the Metropolis concept of establishing a Markov process whose stationary state is the wanted distribution. Instead, it builds weighted instances according to a biased distribution. If the bias is optimal, all weights are equal and importance sampling is perfect. If not, "population control" is applied by cloning/killing configurations with too high/low weight. It uses the fact that nontrivial problems in statistical physics are high dimensional. Therefore, instances are built up in many steps, and the final weight can be guessed at an early stage. In contrast to evolutionary algorithms, the cloning/killing is done such that the wanted distribution is strictly observed without simultaneously keeping a large population in computer memory. We apply this method (which is also closely related to diffusion type quantum Monte Carlo) to several problems of polymer statistics, population dynamics, and percolation.