Timeline for Random noncrossing chords of a circle
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Sep 2, 2010 at 15:16 | comment | added | Louigi Addario-Berry | But there are constants. Together Theorem 1.1 (ii) and Proposition 4.1 imply that, viewing the circle as the unit circle in the complex plane, the expected distance from 1 to $e^{2 \pi i u}$ is about $2.0(n u (1-u))^{\beta}$, where $\beta = (\sqrt{17}-3)/2$. (The 2.0 here is a pretty good approximation to what is in fact a ratio of Gamma functions.) This gives expected distance a little over 12 between $1$ and $-1$, for example. | |
| Sep 2, 2010 at 14:34 | comment | added | Joseph O'Rourke | @louigi: Cool! That evaluates to approx. 13 for $n{=}100$. | |
| Sep 2, 2010 at 14:21 | comment | added | Louigi Addario-Berry | The height of that tree is conjectured (by Nicolas Curien in a talk I saw) to be of the order n^{(sqrt(17)-3)/2} -- if you pick two uniformly random nodes, this will be the rough distance between them. But an argument is missing to show that there is no small exceptional set of nodes at greater distance from one another. | |
| Sep 2, 2010 at 13:55 | comment | added | Joseph O'Rourke | Ah, thanks! I would never have hit upon "random geodesic laminations"! | |
| Sep 2, 2010 at 13:51 | history | answered | Gjergji Zaimi | CC BY-SA 2.5 |