On a surface formed by randomly gluing together polygonal discs

Abstract

Starting with a collection of $n$ oriented polygonal discs, with an even number $N$ of sides in total, we generate a random oriented surface by randomly matching the sides of discs and properly gluing them together. Encoding the surface in a random permutation $\gamma$ of $[N]$, we use the Fourier transform on $S_N$ to show that $\gamma$ is asymptotic to the permutation distributed uniformly on the alternating group $A_N$ ($A_N^c$ resp.) if $N-n$ and $N/2$ are of the same (opposite resp.) parity. We use this to prove a local central limit theorem for the number of vertices on the surface, whence for its Euler characteristic $\chi$. We also show that with high probability the random surface consists of a single component, and thus has a well-defined genus $g=1-\chi/2$, which is asymptotic to a Gaussian random variable, with mean $(N/2-n-\log N)/2$ and variance $(\log N)/2$.