Case studies in omniparametric simulation

Abstract: In the eld of particle systems and growths models simulation is an important tool. When explicit calculations are too complex or impossible to perform we may use simulations instead. We adapt a new technique here denoted omniparametric simulation, to the two-type Richardson, Ising and Potts models. Omniparametric means simulating for all parameter values at the same time giving us something else than ordinary samples, but by xingthe parameter value we can always retrieve an ordinary sample. We use only one dimensional parameters, so for the random cluster and Potts models we x q at some value and consider it known. For the two-type Richardson model we use symmetry and rescale time to eliminate one of the two parameters. We study We study asymmetric simultaneous survival for the twotype Richardson model using omniparametric simulations. The belief is that if both types are equally strong the can survive for all times but if one type is stronger than the other this can not happen. We do not nd any indication of the existence of so called exceptional values < 1 where simultaneous survival may be possible.. We develop a simple test procedure to see how strong the indications against exceptional values are and also which exceptional values tests may rule out, and also consider how large subsets of Z2 we must use. For the Ising and Potts models we use omniparametric simulations to nd smooth estimates of functions for model characteristics such as connection probabilities and susceptbility. The characteristics are then used for parameter estimation, we construct both point estimate and condence intervals. Based on partial observations we develop three methods, two using asymptotic theory, and on non-asymptotic. The method for constructing point estimate are the same for all three approaches, the difference lies in ho we capture the variance of the statistic. We perform extensive testing of the methods and elaborate some on the difference between the model used in simulations and the experienced from data.