On a surface formed by randomly gluing together polygonal discs

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• 作者：Sergei Chmutov ; ^{chmutov@math.ohio-state.edu" class="auth_mail" title="E-mail the corresponding author} ; Boris Pittel¹ ; ^{bgp@math.ohio-state.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C80 ; 05C30 ; 05A16 ; 05E10 ; 34E05 ; 60C05
• 刊名：Advances in Applied Mathematics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：73
• 期：Complete
• 页码：23-42
• 全文大小：486 K

文摘

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]$[N]$, we use the Fourier transform on S_N$S_N$ to show that γ   is asymptotic to the permutation distributed uniformly on the alternating group A_N$A_N$ ($A_N^c$ resp.) if e6707c692bb" title="Click to view the MathML source">N−n$N-n$ and 22e9eeee34ffee0" title="Click to view the MathML source">N/2$N/2$ 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→∞$N\to \infty$, uniformly in n  ) the random surface consists of a single component, and thus has a well-defined genus g=1−χ/2$g=1-\chi /2$, which is asymptotic to a Gaussian random variable, with mean (N/2−n−log⁡N)/2$(N/2-n-\log N)/2$ and variance (log⁡N)/4$(\log N)/4$.