边数过多的随机几何图的局部化
Localization in random geometric graphs with too many edges
摘要：
考虑一个随机几何图$G(χ_n, r_n)$，它是由强度为$n$的Poisson点过程$χ_n$的两个顶点在单位环面上的距离小于参数$r_n$时连接而成的.该模型的条件是罕见的事件，观察到的边缘数，$|E|$，是大于$(1 + δ)\mathbb{E}(|E|)$，对于一些固定的$δ>0$。本文证明了在条件作用下，对于任意给定的$\varepsilon >0$，高概率存在一个直径为$r_n$的球，该球包含一个至少由$\sqrt{2 δ\mathbb{E}(|E|)}(1 - \varepsilon)$顶点组成的团.直观地说，这个区域包含所有的“多余”的边缘，该图是被迫包含的条件事件，直到较低的顺序校正。作为这个结果的一个结果，我们证明了随机几何图的边数的上尾的一个大偏差原理。这个大偏差原理的速率函数原来是非凸的。
英文摘要：
We consider a random geometric graph $G(χ_n, r_n)$, given by connecting two vertices of a Poisson point process $χ_n$ of intensity $n$ on the unit torus whenever their distance is smaller than the parameter $r_n$. The model is conditioned on the rare event that the number of edges observed, $|E|$, is greater than $(1 + δ)\mathbb{E}(|E|)$, for some fixed $δ>0$. This article proves that upon conditioning, with high probability there exists a ball of diameter $r_n$ which contains a clique of at least $\sqrt{2 δ\mathbb{E}(|E|)}(1 - \varepsilon)$ vertices, for any given $\varepsilon >0$. Intuitively, this region contains all the "excess" edges the graph is forced to contain by the conditioning event, up to lower order corrections. As a consequence of this result, we prove a large deviations principle for the upper tail of the edge count of the random geometric graph. The rate function of this large deviation principle turns out to be non-convex.
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