The Expected Genus of a Random Chord Diagram

摘要

To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g _n denote the expected genus of a randomly chosen oriented chord diagram of order n. We show that g _n satisfies: g_n=fracn2-varTheta (lnn).g_n=frac{n}{2}-varTheta (ln n).