Comments on: Random turtles in the hyperbolic plane

By: Danny Calegari

Danny Calegari — Sat, 22 Dec 2012 14:02:00 +0000

Lemonade stand is awesome!

By: galoisrepresentations

galoisrepresentations — Sat, 22 Dec 2012 08:01:43 +0000

I haven’t used turtle graphics/logo since 1986 or so (in a computer “class” which also covered Transylvania and lemonade stand), but I must say I remember it fondly.

By: Danny Calegari

Danny Calegari — Sun, 16 Dec 2012 23:33:45 +0000

Hey Dylan – the random function is bounded on one side of the phase transition locus, so the variance of (the function whose graph is being plotted) is identically zero there. On the other hand, the variance is strictly positive on the other side. Hence the function is not real analytic at any point on the phase transition locus. (But: it is infinitely tangent to the identity there)

Also: I didn’t mean to give the impression that the quasigeodesity was certified by the turtle path being embedded. It just so happens that below the phase transition the paths are embedded and are quasigeodesics, and above the phase transition the paths are not embedded and are not quasigeodesics, with probability one.

By: Dylan Thurston

Dylan Thurston — Sun, 16 Dec 2012 21:49:17 +0000

What makes you sure the graph is not real analytic at the phase transition? The graph is pretty hard to read in the 3-d form like that; a 2-d slice would help.

Also, I must be confused about the definition of quasi-geodesic, because I didn’t think self-intersection was a barrier to being a quasi-geodesic.