TITLE: A random variation on Polya's orchard problem
QUESTION [10 upvotes]: Polya's orchard problem is as follows:

"How thick must the
  trunks of the trees in a regularly spaced circular orchard grow if they are
  to block completely the view from the center?"

See, e.g.,

Allen, Thomas Tracy. "Polya's orchard problem."
  American Mathematical Monthly (1986): 98-104.
  JSTOR link

or this earlier MO question.
This problem has been well-studied. One crude way to phrase what is known
is that, if the trees/disks centered at each lattice point have
radius $r$, the furthest distance one can see from the origin
is $R \approx 1/r$.
Here is my question. Suppose instead of disks centered on each
lattice point, we have a randomly oriented segment (a $1$-dimensional disk)
centered on each lattice point, of length $2r$, i.e., of radius $r$.

Q. Is it still the case that the furthest one can see from the
  origin is expected to be $R=c/r$ for some constant $c$?

Here is an example, with $r=3/8$. The "Polya radius"
is about $2.47$, but in this one random instance,
visibility extends about $3$ times further,
$R \approx 8$:

 


In $\mathbb{R}^3$, the same question can be asked with now $2$-dimensional
randomly oriented disks centered on each lattice point.
And of course the question generalizes to $\mathbb{R}^d$.

REPLY [5 votes]: Probably this problem was not considered so far, but there are some related articles.

"The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems"  by Jens Marklof, Andreas Strömbergsson.
In partiqular they give the number of spheres in a random direction.
"Visibility and directions in quasicrystals"  by Jens Marklof, Andreas Strömbergsson. (In each direction points which are closed to a straight line form a "quasicrystal".)

3."Perfect Retroreflectors and Billiard Dynamics"  by Pavel Bachurin, Konstantin Khanin, Jens Marklof, Alexander Plakhov
See also Marklof's talk at ICM-2014 (lecture I09_04) for an introduction.
I think that this authors can solve your problem in arbitrary dimension.