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Using high-order multistep integration methods optimized to minimize roundoff error propagation, we performed fully three-dimensional integrations of planetesimal trajectories for 100 million years to examine possible niches in the Jupiter/Saturn, Saturn/Uranus, and Uranus/Neptune zones. We computed the trajectories of 100,000 massless particles in the Jupiter/Saturn zone, and 10,000 particles each in the Saturn/Uranus and Uranus/Neptune zones. A planetesimal situated between two large perturbers resides near a separatrix that determines the qualitative dynamical character of its orbit. Given sufficient time and according to the accuracy of the scheme, the dynamical character will artificially change in the vicinity of a separatrix. The modified Cowell-St\"ormer integrator employed in these surveys produces a longitude error for Jupiter bounded below $2^\circ$ in 1 billion years, orders of magnitude smaller than earlier surveys. Other surveys, particularly those based on mapping techniques, sacrificed numerical accuracy in order to permit greater integration times. We believe that this may have resulted in an incorrect assessment of the relative significance of the different physical ejection mechanisms present.
The highly accurate integration of such large numbers of particles (i.e. employing hundreds to thousands of times more particles than in previous surveys) allows us to make statistically significant inferences about the dynamics and dominant physical mechanisms in these regions. Sample sizes should be at least 10,000 so that the ``statistics of small numbers'' is not a factor. In assessing the reliability of such ``counting experiments,'' the relative error (i.e. $3\sigma$ level) is on the order of $3/\sqrt{N}$ for a sample of size $N$---for a survey containing 100,000 particles, this is 1\% of the sample; for 100, particles, counts with fewer than 30\% the sample are not significant. The very large size of our samples, coupled to the increased numerical accuracy in our surveys permits us for the first time to explore the relative significance of the various ejection mechanisms present.
