线性弦图中的气泡：桥与结晶图
Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams
AI总结：
本文研究线性弦图中桥的渐近分布，证明桥数渐近正态，并引入结晶图概念，给出其短弦数的渐近分布及峰值位置。
摘要：
在线性弦图中，短弦连接相邻顶点，而气泡是指不含短弦的区域。我们定义桥为连接气泡内部顶点与气泡外部顶点的弦。基于前期工作，我们研究了大气泡和大图极限下桥的分布，并证明桥的数量是渐近正态的，得到了相应的均值和方差作为气泡大小函数的表达式。我们引入结晶图的概念，其定义标准为图中所有弦要么是短弦，要么是桥。我们按所含短弦的数量对结晶图进行计数，并给出大结晶图极限下的渐近分布。我们证明，对于非常大的图，短弦的数量是正态的，并且在 $\sqrt{2n/\log n}$ 处尖锐地达到峰值，其中 $n$ 是图中弦的总数。
英文摘要：
In a linear chord diagram a short chord joins adjacent vertices while a bubble is a region devoid of short chords. We define a bridge to be a chord joining a vertex interior to a bubble to one exterior to it. Building on earlier work, we investigate the distribution of bridges in the limit of large bubbles and diagrams, and show that the number of bridges is asymptotically normal, obtaining expressions for the associated mean and variance as a function of bubble size. We introduce the notion of a crystallized diagram, defined by the criteria that all its chords are either short or are bridges. We count the number of crystallized diagrams by the number of short chords they contain, and provide the asymptotic distribution in the limit of large crystallized diagrams. We show that for very large diagrams, the number of short chords is normal, and sharply peaked at $\sqrt{2n/\log n}$, where $n$ is the total number of chords in the diagram.
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