Anti-concentration for subgraph counts in random graphs
Anti-concentration for subgraph counts in random graphs
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DOI：
10.1214/20-aop1490
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发表时间：
2019-05
期刊：
影响因子：
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通讯作者：
J. Fox;Matthew Kwan;Lisa Sauermann
中科院分区：
文献类型：
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作者：
J. Fox;Matthew Kwan;Lisa Sauermann
修复图$ h $和（0,1）$中的某些$ p \，让$ x_h $是随机图$ g（n，p）$的$ h $的副本。自Erdős和Renyi的基础工作以来，形式已经进行了深入的研究。到目前为止不理。 $ x_h $在某些小间隔中落下或等于本文的某些特定价值的可能性几乎是最佳的结果，如果连接$ h $，那么对于任何$ x \ in } $我们有$ \ pr（x_h = x）\ le n^{1-v（h）+o（1）} $。 ，并依靠新的抗解不平等行为“几乎线性”的向量。
Fix a graph $H$ and some $p\in (0,1)$, and let $X_H$ be the number of copies of $H$ in a random graph $G(n,p)$. Random variables of this form have been intensively studied since the foundational work of Erdős and Renyi. There has been a great deal of progress over the years on the large-scale behaviour of $X_H$, but the more challenging problem of understanding the small-ball probabilities has remained poorly understood until now. More precisely, how likely can it be that $X_H$ falls in some small interval or is equal to some particular value? In this paper we prove the almost-optimal result that if $H$ is connected then for any $x\in \mathbb{N}$ we have $\Pr(X_H=x)\le n^{1-v(H)+o(1)}$. Our proof proceeds by iteratively breaking $X_H$ into different components which fluctuate at "different scales", and relies on a new anticoncentration inequality for random vectors that behave "almost linearly".