Timeline for The distribution of the shortest path through $n$ points
Current License: CC BY-SA 3.0
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| Oct 7, 2014 at 5:08 | comment | added | Noam D. Elkies | Thanks. Yes, the restriction to ${\cal S}^n$ is one reason (though a typical path of total length $c\sqrt{n}$ has a good chance of staying in $\cal S$ anyway); another is that there may be more than one short path (for starters they always come in pairs because the path can be traversed in either direction), so the probability that there's at least one short path might be significantly less than the expected number of short paths. | |
| Oct 7, 2014 at 5:03 | comment | added | Will Schaefer | That is pretty awesome. So, the fact that this is an upper bound comes from the fact that the region whose volume you take doesn't have to lie in $\mathcal{S}^{n}$? | |
| Oct 7, 2014 at 4:41 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |