The genus of a random chord diagram is asymptotically normal

Abstract

Let Gn be the genus of a two-dimensional surface obtained by gluing, uniformly at random, the sides of an n-gon. Recently Linial and Nowik proved, via an enumerational formula due to Harer and Zagier, that the expected value of Gn is asymptotic to (n−logn)/2 for n→∞. We prove a local limit theorem for the distribution of Gn, which implies that Gn is asymptotically Gaussian, with mean (n−logn)/2 and variance (logn)/4.