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Upper Tail Large Deviations of Regular Subgraph Counts in Erdős-Rényi Graphs in the Full Localized Regime
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2021-12-22 , DOI: 10.1002/cpa.22036
Anirban Basak 1 , Riddhipratim Basu 1
Affiliation  

For a urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0001-regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0002, an Erdős-Rényi graph on n vertices with edge probability p, has generated significant interest. For urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0003 and urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0004, where urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0005 is the number of vertices in H, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of H in urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0006 exceeds its expectation by a constant factor is predicted to hold at a speed urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0007, and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years, covering progressively broader ranges of p, the upper tail large deviations for cliques of fixed size were proved by Harel, Mousset, and Samotij in the entire localized regime. This paper establishes the conjecture for all connected regular graphs in the whole localized regime. © 2021 Wiley Periodicals LLC.

中文翻译:

全局域化条件下 Erdős-Rényi 图中正则子图计数的上尾大偏差

对于urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0001- 规则连通图H确定上尾大偏差的问题对于H的副本数urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0002,一个 Erdős-Rényi 图在n个顶点上,边缘概率为p,引起了人们的极大兴趣。对于urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0003urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0004,其中是Hurn:x-wiley:00103640:media:cpa22036:cpa22036-math-0005中的顶点数,上尾大偏差事件被认为是由于局部结构的存在而发生的。在这种情况下, H in的副本数量超过其预期一个常数因子的大偏差事件预计将保持在一个速度urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0006urn:x-wiley:00103640:media:cpa22036:cpa22036-math-0007,并且推测速率函数由平均场变分问题的解给出。经过近年来的一系列发展,逐渐覆盖更广泛的p范围,Harel、Mousset 和 Samotij 在整个本地化制度中证明了固定大小的集团的上尾大偏差。本文建立了整个局部区域中所有连通正则图的猜想。© 2021 Wiley Periodicals LLC。
更新日期:2021-12-22
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