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Summary: 5 Upper bounds for c(0+)
This is written (November 2006) in the format of an Appendix to the paper
Percolating paths through random points by David Aldous and Maxim
Krikun, published as ALEA 1 (2006) 89-109. The purpose is to show some
simulations and record (for the constant c(0+) in dimension 2)
· a heuristic upper bound of 0.52
· an upper bound of 0.607 which is rigorous up to sampling error in a
fixed-size Monte Carlo simulation.
These are based on a certain "crude algorithm" (described later) for finding
paths with small average edge-length. This algorithm is not claimed to
be close to optimal (developing near-optimal heuristic algorithms for the
problems considered in this paper remains an open problem) but is easy to
implement.
5.1 A heuristic upper bound
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