On a surface formed by randomly gluing together polygonal discs

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 S N to show that γ 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 also for its Euler characteristic ?. 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 - ? / 2 , which is asymptotic to a Gaussian random variable, with mean ( N / 2 - n - log ? N ) / 2 and variance ( log ? N ) / 4 .