论文标题
最少的小气门原理和对随机图的应用
The fewest-big-jumps principle and an application to random graphs
论文作者
论文摘要
我们证明了n个独立的重尾随机变量之和的大偏差原理，该变量在位置n处受到移动的截止边界的约束。以规模为n的总和为大的条件，我们表明有限数量的总数在截止边界附近具有值，而其余变量仍然遵守大量法律。这将概括为随机变量的众所周知的单高跳跃原理，而不会截止仅发生必要的跳跃数量的情况。作为一个应用程序，我们考虑了一个随机图，该图具有由带有侧面2n+1的圆环的晶格点给出的顶点集。每个顶点都是从重尾分布中采样随机半径的球的中心。从中央顶点绘制了方向的边缘到该球中的所有其他顶点。当该图以具有异常数量的边缘为条件时，我们使用主要结果表明，随着n的无穷大，过量的超级成分凝结在固定数量有限的宏观射击量的随机散布顶点中。相比之下，对顶点的固定体不会发生凝结，这一切都保持了微观的大小。
We prove a large deviation principle for the sum of n independent heavy-tailed random variables, which are subject to a moving cut-off boundary at location n. Conditional on the sum being large at scale n, we show that a finite number of summands take values near the cut-off boundary, while the remaining variables still obey the law of large numbers. This generalises the well-known single-big-jump principle for random variables without cut-off to a situation where just the minimal necessary number of jumps occur. As an application, we consider a random graph with vertex set given by the lattice points of a torus with sidelength 2N+1. Every vertex is the centre of a ball with random radius sampled from a heavy-tailed distribution. Oriented edges are drawn from the central vertex to all other vertices in this ball. When this graph is conditioned on having an exceptionally large number of edges we use our main result to show that, as N goes to infinity, the excess outdegrees condense in a fixed, finite number of randomly scattered vertices of macroscopic outdegree. By contrast, no condensation occurs for the indegrees of the vertices, which all remain microscopic in size.
