Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $[n]:=\{1,\dots ,n\}$ with $m=m(n)$ edges. We show that in the sparse regime, when $m/n\le 1$, with high probability the maximum degree of $P(n,m)$ takes at most two different values. In contrast, this is not true anymore in the dense regime, when $m/n>1$, where the maximum degree of $P(n,m)$ is not concentrated on any subset of $[n]$ with bounded size.

中文翻译：

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随机平面图中最大度数的集中

让$磷(n,米)$是从顶点集上所有平面图的类中均匀随机选择的一个图$[n]：=\{1,\dots ,n\}$和$米=米(n)$边缘。我们证明了在稀疏状态下，当$米/n\le 1$, 最大程度的概率很高$磷(n,米)$最多取两个不同的值。相反，这在稠密体制中不再正确，当$米/n>1$, 其中最大程度$磷(n,米)$不集中在任何子集上$[n]$有界的大小。