3,189 research outputs found
Fast simulation of large-scale growth models
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.Comment: 27 pages, 9 figures. To appear in Random Structures & Algorithm
The two-sample problem for Poisson processes: adaptive tests with a non-asymptotic wild bootstrap approach
Considering two independent Poisson processes, we address the question of
testing equality of their respective intensities. We first propose single tests
whose test statistics are U-statistics based on general kernel functions. The
corresponding critical values are constructed from a non-asymptotic wild
bootstrap approach, leading to level \alpha tests. Various choices for the
kernel functions are possible, including projection, approximation or
reproducing kernels. In this last case, we obtain a parametric rate of testing
for a weak metric defined in the RKHS associated with the considered
reproducing kernel. Then we introduce, in the other cases, an aggregation
procedure, which allows us to import ideas coming from model selection,
thresholding and/or approximation kernels adaptive estimation. The resulting
multiple tests are proved to be of level \alpha, and to satisfy non-asymptotic
oracle type conditions for the classical L2-norm. From these conditions, we
deduce that they are adaptive in the minimax sense over a large variety of
classes of alternatives based on classical and weak Besov bodies in the
univariate case, but also Sobolev and anisotropic Nikol'skii-Besov balls in the
multivariate case
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