AI 中文总结
本文针对 Erdős-Rényi 图在 $Np \gg (\log N)^2$ 区域，利用新提出的 Bernoulli 流方法证明体相特征向量最优离域化及局部谱普适性。
AI 中文摘要
我们研究 Erdős-Rényi 图 $\mathbb G(N,p)$ 的邻接矩阵 $A$ 在 $Np \gg (\log N)^2$ 区域内的特征值和特征向量。我们建立了体相特征向量 $\boldsymbol u$ 的最优各向同性离域化，即对于任何确定性的归一化向量 $\boldsymbol v$，以极高概率有 $\langle \boldsymbol v, \boldsymbol u\rangle^2 \leq \frac{C \log N}{N}$。此外，我们通过证明局部谱统计量与 GOE 的统计量一致，证明了体相中的局部谱普适性。我们证明的主要工具是 $A$ 的预解式的局部律，该局部律达到最优谱尺度并具有最优误差界。其证明依赖于一种我们称之为 Bernoulli 流的新方法，这是一种特征流方法，其中通常的布朗过程被 Bernoulli 过程取代，图的每条边形成一个独立的马尔可夫过程，以单位速率从闭合状态跳转到开放状态。谱参数根据适当构造的矩阵值 Bernoulli 特征流流动。
英文摘要
We study the eigenvalues and eigenvectors of the adjacency matrix $A$ of the Erdős-Rényi graph $\mathbb G(N,p)$ in the regime $Np \gg (\log N)^2$. We establish optimal isotropic delocalization for the bulk eigenvectors $\boldsymbol u$, meaning that $\langle \boldsymbol v, \boldsymbol u\rangle^2 \leq \frac{C \log N}{N}$ with very high probability for any deterministic normalized $\boldsymbol v$. In addition, we prove local spectral universality in the bulk by showing that the local spectral statistics coincide with those of the GOE. The main tool of our proof is a local law for the resolvent of $A$, down to optimal spectral scales and with optimal error bounds. Its proof relies on a new approach to local laws that we call the Bernoulli flow. It is a characteristic flow method in which the usual Brownian process is replaced by a Bernoulli process, where each edge of the graph forms an independent Markov process that jumps at unit rate from closed to open. The spectral parameter flows according to a suitably constructed matrix-valued Bernoulli characteristic flow.
