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Case studies in omniparametric simulation

Doctoral thesis, 2006

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.

omnithermal simulation

omniparametric simulationpercolatiion

simulation driven parameter estimation

parameter estimation

random cluster model

partial observations

growth model

Potts model

two-type Richardson model

Markov chain

Ising model

Richardson model