The genus of a random chord diagram is asymptotically normal

  • Authors:

  • Sergei Chmutov;Boris Pittel

  • Affiliations:

  • Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, OH 43210, United States;Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, OH 43210, United States

  • Venue:
  • Journal of Combinatorial Theory Series A
  • Year:
  • 2013

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Abstract

Let G"n 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 G"n is asymptotic to (n-logn)/2 for n-~. We prove a local limit theorem for the distribution of G"n, which implies that G"n is asymptotically Gaussian, with mean (n-logn)/2 and variance (logn)/4.