  Oriented loops on an orientable surface are, up to equivalence by free
homotopy, in one-to-one correspondence with the conjugacy classes of the
surface's fundamental group. These conjugacy classes can be expressed (not
uniquely in the case of closed surfaces) as a cyclic word of minimal length in
terms of the fundamental group's generators. The self-intersection number of a
conjugacy class is the minimal number of transverse self-intersections of
representatives of the class. By using Markov chains to encapsulate the
exponential mixing of the geodesic flow and achieve sufficient independence, we
can use a form of the central limit theorem to describe the statistical nature
of the self-intersection number. For a class chosen at random among all classes
of length n, the distribution of the self intersection number approaches a
Gaussian when n is large. This theorem generalizes the result of Steven Lalley
and Moira Chas to include the case of closed surfaces.
