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

TL;DR

This paper proves delocalization, quantum ergodicity, and universality of eigenvalue statistics for high-dimensional random band matrices and related models when the bandwidth exceeds a polynomial scale, revealing a localization-delocalization transition.

Contribution

It establishes delocalization and universality results for random band matrices and block models in dimensions $d \\ge 3$ with bandwidth above a polynomial threshold, extending previous understanding.

Findings

Bulk eigenvectors are delocalized for large bandwidth W.

Quantum unique ergodicity (QUE) holds in the bulk.

Bulk eigenvalue statistics are universal in the large-N limit.

Abstract

We study $N \times N$ random band matrices $H = (H_{xy})$ 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 satisfies $\mathbb{E}|H_{xy}|^2 = S_{xy}$, with $S_{xy} = 0$ whenever the distance between $x$ and $y$ exceeds a bandwidth parameter $W$. We prove that if $W \geq N^{\mathfrak{c}}$ for some constant $\mathfrak{c} > 0$, then in the large-$N$ limit, bulk eigenvectors are delocalized, quantum unique ergodicity (QUE) holds, and the local bulk eigenvalue statistics are universal. Our proof is based on the tree approximation of the loop hierarchy (arXiv:2501.01718) and diagrammatic techniques developed in earlier works (arXiv:1807.02447, arXiv:2104.12048, arXiv:2107.05795, arXiv:2412.15207, arXiv:2503.07606). Besides random band matrices, we also study two classical non-mean-field random matrix models: the Wegner orbital and the block Anderson models. Specifically, we consider Hermitian matrices $H = V + g \Psi$ on the same discrete torus $(\mathbb{Z} / \sqrt[d]{N} \mathbb{Z})^d$, where $V$ is a random block potential consisting of i.i.d. complex Gaussian diagonal blocks of size $W^d \times W^d$, and $\Psi$ encodes the interactions between neighboring blocks--random in the Wegner orbital model and deterministic in the block Anderson model. The parameter $g > 0$ represents the coupling strength between blocks. Assuming again that $W \geq N^{\mathfrak{c}}$, we establish delocalization of bulk eigenvectors, QUE, and bulk universality under the condition $W^{-d/2+\varepsilon}\le g \le \varepsilon^{-1}$ for any small constant $\varepsilon>0$. Combined with the localization results of arXiv:1608.02922 for $g \ll W^{-d/2}$, this identifies a localization--delocalization transition at the scale $g=W^{-d/2}$ in dimensions $d \ge 3$.