TITLE: Path length of ball on tilted, perforated plane
QUESTION [10 upvotes]: Imagine that an $\epsilon$-radius hole is punched in the plane centered
on every integer-coordinate point.
Now a point "ball" is dropped on the plane at a random spot $p$.
If $p$ has not already dropped through a hole, 
the plane is tilted in a random direction,
and the ball rolls along the gradient vector $u$.

What is the expected length $L(\epsilon)$ of its roll before it falls through a hole?

If $p$ is in a hole upon generation, then $L=0$.
Otherwise $L$ is determined by the first encounter with a hole,
i.e., the ray from $p$ along direction $u$ passes within distance
$\epsilon$ of some hole center $c$.
For example, here $\epsilon=\frac{1}{4}$ and $c=(2,1)$, with $L
\approx 1.6$:
        

I am particularly interested in the growth rate of $L(\epsilon)$.
Intuition from another viewpoint suggests $\sim \frac{1}{\epsilon}$.
Shrink the holes to points, and grow the ball to an $\epsilon$-radius
disk.  This sweeps out a rectangle of area $2 \epsilon L$
for a roll of length $L$.  Very crudely, when this
area reaches $1$ (or $1-\pi \epsilon^2$), I would expect it to include a lattice point.
So perhaps $L \approx \frac{1}{2\epsilon}$.  But this reasoning is
surely not sound.
Perhaps there is an approach that employs techniques from rational approximations?
All ideas welcomed—Thanks!
Addendum.  Here are two histograms for $L(\epsilon)$ with $\epsilon=\frac{1}{4}$.
The first is of 200 random points, the second of 300 random points.
        
They begin to illustrate Doug Zare's point that very long paths occur not infrequently.
In the 300-point example, one path has $L \approx 63$, while more than half the paths
have $L \le 1$. The probability that $L \le 10$ is $>98$%, even though the expected
length is $\infty$!

REPLY [4 votes]: J. Marklof and A. Strömbergsson have pretty much complete results for the distribution you're interested in - both on the plane and in higher dimensions. Check 
J. Marklof and A. Strömbergsson, The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems, Annals of Mathematics 172 (2010) 1949–2033 
J. Marklof and A. Strömbergsson, The Boltzmann-Grad limit of the periodic Lorentz gas, Annals of Mathematics 174 (2011) 225-298