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 by a random permutation γ of [N], we use the Fourier transform on SN to show that γ is asymptotic to the permutation distributed uniformly on the alternating group AN (ANc 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 also for its Euler characteristic χ. We also show that with high probability (as N→∞, uniformly in n) the random surface consists of a single component, and thus has a well-defined genus g=1−χ/2, which is asymptotic to a Gaussian random variable, with mean (N/2−n−log⁡N)/2 and variance (log⁡N)/4.