 
Summary: The Complexity of the Outer Face in Arrangements of
Random Segments
Noga Alon Dan Halperin Oren Nechushtan Micha Sharir
School of Computer Science
TelAviv University, Israel
{nogaa,danha,theoren,michas}@post.tau.ac.il
ABSTRACT
We investigate the complexity of the outer face in arrange
ments of line segments of a fixed length in the plane, drawn
uniformly at random within a square. We derive upper
bounds on the expected complexity of the outer face, and es
tablish a certain phase transition phenomenon during which
the expected complexity of the outer face drops sharply as
a function of the total number of segments. In particular
we show that up till the phase transition the complexity of
the outer face is almost linear in n, and that after the phase
transition, the complexity of the outer face is roughly pro
portional to
