SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1069-1083, January 2020.
The upper tail problem for the largest eigenvalue of the Erdös--Rényi random graph $\mathcal{G}_{n,p}$ is to estimate the probability that the largest eigenvalue of the adjacency matrix of $\mathcal{G}_{n,p}$ exceeds its typical value by a factor of $1+\delta$. In this note we show that for $\delta >0$ fixed, and $p \rightarrow 0$ such that $n^{\frac{1}{2}} p \rightarrow \infty$, the upper tail probability for the largest eigenvalue of $\mathcal{G}_{n,p}$ is $\exp[-(1+o(1)) \min\{\tfrac{(1+\delta)^2}{2}, \delta(1+\delta) \} n^{2}p^{2}\log (1/p)].$ In the same regime of $p$, we show that the second largest eigenvalue $\lambda_2(\mathcal{G}_{n,p})$ of the adjacency matrix of $\mathcal{G}_{n,p}$ satisfies $\mathbb{P}(\lambda_2(\mathcal{G}_{n,p})\ge \delta np) = \exp[-(1+o(1)) \tfrac{1}{2} \delta^2n^2p^2 \log (1/p)],$ where $\delta =\delta_n < 1$ can depend on $n$ such that $\delta n^{\frac{1}{2}} p \rightarrow \infty$, which covers deviations of $\lambda_2(\mathcal{G}_{n,p})$ between $n^{\frac{1}{2}}$ and $np$. Our arguments build on recent results on the large deviations of the largest eigenvalue and related nonlinear functions of the adjacency matrix in terms of natural mean-field entropic variational problems.

中文翻译：

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随机图的边缘特征值的上尾

SIAM离散数学杂志，第34卷，第2期，第1069-1083页，2020年1月。
Erdös-Rényi随机图$ \ mathcal {G} _ {n，p} $的最大特征值的上尾问题是估计$ \ mathcal {G} _邻接矩阵的最大特征值的概率{n，p} $比其典型值高出$ 1 + \ delta $。在此注释中，我们显示了对于$ \ delta> 0 $固定的，而$ p \ rightarrow 0 $这样的$ n ^ {\ frac {1} {2}} p \ rightarrow \ infty $， $ \ mathcal {G} _ {n，p} $的最大特征值是$ \ exp [-（1 + o（1））\ min \ {\ tfrac {（1+ \ delta）^ 2} {2}， \ delta（1+ \ delta）\} n ^ {2} p ^ {2} \ log（1 / p）]。$在相同的$ p $体制中，我们显示出第二大特征值$ \ lambda_2（ $ \ mathcal {G} _ {n，p} $的邻接矩阵的\ mathcal {G} _ {n，p}）$满足$ \ mathbb {P}（\ lambda_2（\ mathcal {G} _ {n ，p}）\ ge \ delta np）= \ exp [-（1 + o（1））\ tfrac {1} {2} \ delta ^ 2n ^ 2p ^ 2 \ log（1 / p）]，$其中$ \ delta = \ delta_n < 1 $可以取决于$ n $，从而使$ \ delta n ^ {\ frac {1} {2}} p \ rightarrow \ infty $，其中包括$ \ lambda_2（\ mathcal {G} _ {n，p }）$在$ n ^ {\ frac {1} {2}} $和$ np $之间。我们的论据建立在最近结果的基础上，即关于自然均值场熵变分问题的最大特征值和邻接矩阵的相关非线性函数的大偏差。