论文标题
关键Erdős-rényi图的离域转变
Delocalization transition for critical Erdős-Rényi graphs
论文作者
论文摘要
我们分析了关键的erdős-rényi图$ \ mathbb g（n，d/n）$的邻接矩阵的特征向量，其中$ d $是订单$ \ log n $。我们表明，其频谱分为两个阶段：在频谱中间的一个离域相，在该频谱中间，特征向量被完全取代，在光谱边缘附近的半定位相位，在该频谱边缘，特征向量基本上是在少量的顶点上定位的。在半定位的相中，特征向量的质量集中在围绕谐振顶点的少数不相交球中，在每个谐音的顶点中，它都是径向指数衰减的函数。各个阶段之间的过渡很清晰，并且以eigenVector $ \ mathbf w $的定位指数$γ（\ mathbf w）的不连续性表现出来，这是通过$ \ | \ | \ mathbf w \ | _ _ \ | _ \ | _ \ | \ | \ |我们的结果在整个最佳制度$ \ sqrt {\ log n} \ ll d \ leq o（\ log n）$中保持有效。
We analyse the eigenvectors of the adjacency matrix of a critical Erdős-Rényi graph $\mathbb G(N,d/N)$, where $d$ is of order $\log N$. We show that its spectrum splits into two phases: a delocalized phase in the middle of the spectrum, where the eigenvectors are completely delocalized, and a semilocalized phase near the edges of the spectrum, where the eigenvectors are essentially localized on a small number of vertices. In the semilocalized phase the mass of an eigenvector is concentrated in a small number of disjoint balls centred around resonant vertices, in each of which it is a radial exponentially decaying function. The transition between the phases is sharp and is manifested in a discontinuity in the localization exponent $γ(\mathbf w)$ of an eigenvector $\mathbf w$, defined through $\|\mathbf w\|_\infty / \|\mathbf w\|_2 = N^{-γ(\mathbf w)}$. Our results remain valid throughout the optimal regime $\sqrt{\log N} \ll d \leq O(\log N)$.
