Q:

When sticks fall, will they weave?

Imagine $n$ $z$-vertical sticks uniformly spaced around a unit-radius circle in the $xy$-plane.
At $t{=}0$, each is randomly $\epsilon$-perturbed from the vertical, and they fall under
the influence of gravity. Will some sticks form a "teepee" suspended above the $xy$-plane?
Let us assume the sticks are one-dimensional segments of height $h$, perhaps $h{=}2$ so they
span the diameter,
and that their base points are pinned to the plane via universal joints.
It seems possible that a subset of sticks could fall to form a weaving with a cyclic
on-top-of graph, as illustrated below.

Assuming a sufficient coefficient of friction $\mu$ between pairs of sticks, it seems conceivable that such a
structure would not collapse to the plane.
What is your intuition here?  For sufficiently large $n$, and adequate $h$ and $\mu$,
would some sticks form a woven structure above the plane?
Or would all sticks ultimately flatten to the plane?
If the latter, are there natural conditions that would lead to formation of a "teepee"?
I am less interested in probability calculations than in a qualitative assessment.
I've tagged this 'recreational' because it is only a peripheral spinoff of my research.
Addendum.  It is now clear, from the comments of Scott Morrison and Rahul Narain, that
the random perturbations should be random both in direction and magnitude.
Otherwise, as Scott incisively observed, the sticks, with high probability, all fall without touching
one another until they reach the plane (which to me is already rather remarkable!).

A:

Couldn't accomplish this in a comment to Rahul's #1, but here is a physical realization of my drawing with colored pencils. Compare:

The pencils are in contact, but the lead inside could serve as lines in space without contacts.
I don't know how to reconcile this with your quote from Whiteley...

A:

Something doesn't seem right here. Let's ignore the 'coefficient of friction' and just make the sticks out of caramel: if they even touch they glom together. Nevertheless, everything is always flat!
If sticks ever glom, there's a pair of sticks that glom first. However, if you think about a pair of sticks falling in this way (I'm assuming $\epsilon$ is small, so all sticks make the same angle to the horizontal at any moment before the first glom), it's clear that they only actually make contact with each other at the instant they hit the ground. (The only other possibility is having the tips collide, which almost surely never happens.)
Perhaps you can make the sticks thicker and have something happen, but that seems extremely complicated.

