From:

Subject:

I wonder about the validity of your Monte Carlo analysis. It seems

to be based on an intuition about how fast the number of configurations

falls off with the distance from SOLVED. I share the intuition, but

I'm not sure I can rigorize it, and that makes me cautious.

What prevents a group from having a "pointy tail", that is, a "corridor"

of elements at increasing distances from the identity? In fact, does

the number of elements as a function of distance have to be unimodal?

Could this function have a "waist"? Intuitively, this sounds

impossible, but I am wondering what constraints on such functions are known.