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

TL;DR

The paper establishes delocalization, quantum unique ergodicity, and bulk universality for non-mean-field random matrices in dimensions d ≥ 3, focusing on random band matrices and Wegner-type block models. It develops a stochastic flow and loop hierarchy, and proves that the tree-approximation (K-loops) accurately captures the leading dynamics, enabling diffusion-like quantum transport in the bulk under bandwidth W ≳ N^ε and coupling g within a controlled window. A combination of sharp local laws, Ward identities, and diagrammatic methods yield 2-loop and higher-loop bounds, culminating in QUE and bulk universality, with a localization–delocalization transition identified in the block Anderson/Wegner models at g ∼ W^{-d/2}. The results extend the understanding of localization phenomena from 1D/2D to higher dimensions and provide a robust framework for analyzing non-mean-field matrix ensembles through loop-contraction and stochastic flow techniques. The work has potential implications for quantum diffusion in disordered media and for rigorous universality results in non-mean-field random matrix theory.

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 $H = V + g Ψ$ 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 $Ψ$ 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 $d \ge 3$.

Figures (6)

• Figure 1: Possible expansions of \ref{['eq:p=2graph']} and the corresponding molecular graph. The $\times$-dotted edges in the first graph have been omitted.
• Figure 2: Illustration of the construction of paths $\mathfrak{P}_1'$ and $\mathfrak{P}_2'$ (right panel) from $\mathfrak{P}_1$ and $\mathfrak{P}_2$ (left panel).
• Figure 3: Illustration of the spanning trees in case (i) (left panel) and case (iii) (right panel). Blue edges represent external edges connected to external vertices (here blue is used solely for illustration and does not indicate the $+$ charge of $G$-edges as in \ref{['def_graph1']}); purple edges represent the solid edges between $\alpha_1$ and $\alpha_2$; and black edges represent the spanning tree on the internal vertices.
• Figure 4: Illustration of the path-preserving phenomenon in applying \ref{['eq:key_T_reudce']}.
• Figure 5: Replacing $b_i$ with an $M$-loop with 5 sides.

Theorems & Definitions (154)

• Theorem 2.1: Delocalizaiton
• Theorem 2.2: Local semicircle law
• proof : Proof of Theorems \ref{['MR:decol']}
• Theorem 2.3: Quantum unique ergodicity
• Theorem 2.4: Bulk universality
• proof : Proof of Theorem \ref{['MR:QUE']}
• proof : Proof of Theorem \ref{['Thm: B_Univ']}
• Remark 2.6
• Theorem 2.7: Main results for the block Anderson model
• Definition 2.8: Flow framework