标题： 非平均场随机矩阵在维度$d\ge 3$的离域化 显示英文标题

标题： Delocalization of Non-Mean-Field Random Matrices in Dimensions $d\ge 3$

摘要： Consider an $N \times N$ random band matrix $H = \left(H_{xy}\right)$ with mean-zero complex Gaussian entries, where $x, y$ lie on the discrete torus $(\mathbb{Z} / \sqrt[d]{N} \mathbb{Z})^d$ in dimensions $d \ge 3$. 方差轮廓$\mathbb{E}\left|H_{xy}\right|^2 = S_{xy}$在$x$与$y$之间的距离超过给定的带宽参数$W$时消失。 我们证明，如果带宽满足$W \geq N^{\mathfrak{c}}$对某个常数$\mathfrak{c} > 0$，那么在大$N$极限下，体态向量的扩展化和局部半圆律到最优光谱尺度$\eta \ge N^{-1+\varepsilon}$都以高概率成立。 我们的证明基于环层次结构的原始近似（arXiv:2501.01718），并建立在自2021年以来在此系列中开发的方法之上（arXiv:1807.02447, arXiv:2104.12048, arXiv:2107.05795, arXiv:2412.15207, arXiv:2501.01718, arXiv:2503.07606）。 显示英文摘要

摘要： Consider an $N \times N$ random band matrix $H = \left(H_{xy}\right)$ with mean-zero complex Gaussian entries, where $x, y$ lie on the discrete torus $(\mathbb{Z} / \sqrt[d]{N} \mathbb{Z})^d$ in dimensions $d \ge 3$. The variance profile $\mathbb{E}\left|H_{xy}\right|^2 = S_{xy}$ vanishes when the distance between $x$ and $y$ exceeds a given bandwidth parameter $W$. We prove that if the bandwidth satisfies $W \geq N^{\mathfrak{c}}$ for some constant $\mathfrak{c} > 0$, then in the large $N$ limit, both the delocalization of bulk eigenvectors and a local semicircle law down to optimal spectral scales $\eta \ge N^{-1+\varepsilon}$ hold with high probability. Our proof is based on the primitive approximation of the loop hierarchy (arXiv:2501.01718), and builds upon methods which were developed in this series since 2021 (arXiv:1807.02447, arXiv:2104.12048, arXiv:2107.05795, arXiv:2412.15207, arXiv:2501.01718, arXiv:2503.07606).