2014/05/30 by Matthew Wroten, Wroten, Matthew
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Dynamical Systems (math.DS) #FOS: Mathematics #Stochastic processes and statistical mechanics #math.DS
paper · pdf · doi:10.48550/arxiv.1405.7951
arxiv created 2014/05/30 · openalex publication_date 2014/05/30 · arxiv updated 2014/06/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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.