Statistics of Intersections of Curves on Surfaces

Each orientable surface with nonempty boundary can be associated with a planar model, whose edges can then be labeled with letters that read out a surface word. Then, the curve word of a free homotopy class of closed curves on a surface is the minimal sequence of edges of the planar model through which a curve in the class passes. The length of a class of curves is defined to be the number of letters in its curve word. We fix a surface and its corresponding planar model. Fix a free homotopy class of curves ω on the surface. For another class of curves c, let i(ω, c) be the minimal number of intersections of curves in ω and c. In this paper, we show that the mean of the distribution of i(ω, c), for random curve c of length n, grows proportionally with n and appraoches μ(ω) ·n for a constant μ(ω). We also give an algorithm to compute μ(ω) and have written a program that calculates μ(ω) for any curve ω on any surface. In addition, we prove that i(ω, c) approahces a Gaussian distribution as n→∞ by viewing the generation of a random curve as a Markov Chain.