1,334 research outputs found
Continuum percolation for Cox point processes
We investigate continuum percolation for Cox point processes, that is,
Poisson point processes driven by random intensity measures. First, we derive
sufficient conditions for the existence of non-trivial sub- and super-critical
percolation regimes based on the notion of stabilization. Second, we give
asymptotic expressions for the percolation probability in large-radius,
high-density and coupled regimes. In some regimes, we find universality,
whereas in others, a sensitive dependence on the underlying random intensity
measure survives.Comment: 21 pages, 5 figure
Approximation of length minimization problems among compact connected sets
In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some
classical minimization problems involving the length of an unknown
one-dimensional set, with an additional connectedness constraint, in dimension
two. We introduce a term of new type relying on a weighted geodesic distance
that forces the minimizers to be connected at the limit. We apply this approach
to approximate the so-called Steiner Problem, but also the average distance
problem, and finally a problem relying on the p-compliance energy. The proof of
convergence of the approximating functional, which is stated in terms of
Gamma-convergence relies on technical tools from geometric measure theory, as
for instance a uniform lower bound for a sort of average directional Minkowski
content of a family of compact connected sets
Fast and accurate computation of the logarithmic capacity of compact sets
We present a numerical method for computing the logarithmic capacity of
compact subsets of , which are bounded by Jordan curves and have
finitely connected complement. The subsets may have several components and need
not have any special symmetry. The method relies on the conformal map onto
lemniscatic domains and, computationally, on the solution of a boundary
integral equation with the Neumann kernel. Our numerical examples indicate that
the method is fast and accurate. We apply it to give an estimate of the
logarithmic capacity of the Cantor middle third set and generalizations of it
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