稀疏随机图谱中零点扩展态的出现
Emergence of extended states at zero in the spectrum of sparse random graphs
摘要：
我们证实了长期以来的预测，$c=e\approx 2.718$是一个非零的绝对连续的部分（扩展状态）在零的极限谱的Erdens-Renyi随机图的平均度$c$的出现的阈值。这是通过对奇点$z=0$附近的预解式$(A-z)^{-1}$进行详细的二阶分析来实现的，其中$A$是具有平均后代$c$的Poisson-Galton-Watson树的邻接算子。更一般地说，我们的方法适用于任意的么模Galton-Watson树，产生明确的标准，在零处的各种随机图模型的限制谱测度的扩展状态的存在或不存在，在底层的度分布。
英文摘要：
We confirm the long-standing prediction that $c=e\approx 2.718$ is the threshold for the emergence of a non-vanishing absolutely continuous part (extended states) at zero in the limiting spectrum of the Erdős-Renyi random graph with average degree $c$. This is achieved by a detailed second-order analysis of the resolvent $(A-z)^{-1}$ near the singular point $z=0$, where $A$ is the adjacency operator of the Poisson-Galton-Watson tree with mean offspring $c$. More generally, our method applies to arbitrary unimodular Galton-Watson trees, yielding explicit criteria for the presence or absence of extended states at zero in the limiting spectral measure of a variety of random graph models, in terms of the underlying degree distribution.
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