无约束亏格中随机曲面的普适性
Universality for random surfaces in unconstrained genus
摘要：
从一个任意序列的多边形开始，其总周长是$2n$，我们可以通过以统一的方式配对它们的边来构建（定向）曲面。Chmutov和Pittel（arXiv：1503.01816）已经证明，不管我们开始时的多边形的配置如何，以这种方式获得的图的度序列在全变差距离上是显著恒定的，并且朝着Poisson-Dirichlet划分收敛为$n \to \infty$。我们实际上表明，其他几个几何性质的图是普遍的。途中，我们提供了另一种证明的一个弱版本的结果Chmutov和Pittel使用概率技术和有关的思想圈周围的剥离过程的随机平面地图。在这种情况下，我们也填补了现有文献中的空白，调查的性质均匀随机映射的$n$边缘。特别是，我们证明了$n$边随机映射的直径收敛于一个随机变量的法律只采取值在$\{2,3\}$。
英文摘要：
Starting from an arbitrary sequence of polygons whose total perimeter is $2n$, we can build an (oriented) surface by pairing their sides in a uniform fashion. Chmutov and Pittel (arXiv:1503.01816) have shown that, regardless of the configuration of polygons we started with, the degree sequence of the graph obtained this way is remarkably constant in total variation distance and converges towards a Poisson--Dirichlet partition as $n \to \infty$. We actually show that several other geometric properties of the graph are universal. En route we provide an alternative proof of a weak version of the result of Chmutov and Pittel using probabilistic techniques and related to the circle of ideas around the peeling process of random planar maps. At this occasion we also fill a gap in the existing literature by surveying the properties of a uniform random map with $n$ edges. In particular we show that the diameter of a random map with $n$ edges converges in law towards a random variable taking only values in $\{2,3\}$.
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