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Delocalization and quantum diffusion of random band matrices in high dimensions I: Self-energy renormalization

Fan Yang, Horng-Tzer Yau, Jun Yin

TL;DR

The authors address delocalization and quantum diffusion for high-dimensional random band matrices by developing a rigorous, high-order T-expansion with self-energies. Their method translates resolvent behavior into a diffusive kernel through a renormalized diffusion matrix $\Theta$ and systematic self-energy corrections, enabling precise local laws and diffusion-type asymptotics for $\mathbb{E}|G_{xy}(z)|^2$. The framework, built from detailed graph operations and a doubly connected structure, yields delocalization results for bulk eigenvectors in $d\ge 8$ under a mild band-width condition and confirms a quadratic-in-$p$ diffusion form for the Fourier transform of the Green’s function squared. The approach is robust to Gaussian assumptions and is expected to extend to non-Gaussian band matrices, offering a powerful toolkit for non-mean-field disordered systems and connecting spectral properties to quantum diffusion on finite lattices.

Abstract

We consider Hermitian random band matrices $H=(h_{xy})$ on the $d$-dimensional lattice $(\mathbb Z/L\mathbb Z)^d$. The entries $h_{xy}$ are independent (up to Hermitian conditions) centered complex Gaussian random variables with variances $s_{xy}=\mathbb E|h_{xy}|^2$. The variance matrix $S=(s_{xy})$ has a banded structure so that $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In dimensions $d\ge 8$, we prove that, as long as $W\ge L^ε$ for a small constant $ε>0$, with high probability most bulk eigenvectors of $H$ are delocalized in the sense that their localization lengths are comparable to $L$. Denote by $G(z)=(H-z)^{-1}$ the Green's function of the band matrix. For ${\mathrm Im}\, z\gg W^2/L^2$, we also prove a widely used criterion in physics for quantum diffusion of this model, namely, the leading term in the Fourier transform of $\mathbb E|G_{xy}(z)|^2$ with respect to $x-y$ is of the form $({\mathrm Im}\, z + a(p))^{-1}$ for some $a(p)$ quadratic in $p$, where $p$ is the Fourier variable. Our method is based on an expansion of $T_{xy}=|m|^2 \sum_αs_{xα}|G_{αy}|^2$ and it requires a self-energy renormalization up to error $W^{-K}$ for any large constant $K$ independent of $W$ and $L$. We expect that this method can be extended to non-Gaussian band matrices.

Delocalization and quantum diffusion of random band matrices in high dimensions I: Self-energy renormalization

TL;DR

The authors address delocalization and quantum diffusion for high-dimensional random band matrices by developing a rigorous, high-order T-expansion with self-energies. Their method translates resolvent behavior into a diffusive kernel through a renormalized diffusion matrix and systematic self-energy corrections, enabling precise local laws and diffusion-type asymptotics for . The framework, built from detailed graph operations and a doubly connected structure, yields delocalization results for bulk eigenvectors in under a mild band-width condition and confirms a quadratic-in- diffusion form for the Fourier transform of the Green’s function squared. The approach is robust to Gaussian assumptions and is expected to extend to non-Gaussian band matrices, offering a powerful toolkit for non-mean-field disordered systems and connecting spectral properties to quantum diffusion on finite lattices.

Abstract

We consider Hermitian random band matrices on the -dimensional lattice . The entries are independent (up to Hermitian conditions) centered complex Gaussian random variables with variances . The variance matrix has a banded structure so that is negligible if exceeds the band width . In dimensions , we prove that, as long as for a small constant , with high probability most bulk eigenvectors of are delocalized in the sense that their localization lengths are comparable to . Denote by the Green's function of the band matrix. For , we also prove a widely used criterion in physics for quantum diffusion of this model, namely, the leading term in the Fourier transform of with respect to is of the form for some quadratic in , where is the Fourier variable. Our method is based on an expansion of and it requires a self-energy renormalization up to error for any large constant independent of and . We expect that this method can be extended to non-Gaussian band matrices.

Paper Structure

This paper contains 41 sections, 48 theorems, 439 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Random band matrices
  3. Delocalization and local law
  4. Quantum diffusion
  5. $T$-expansion
  6. $T$-expansion and $\text{self-energies}$
  7. Second order $T$-expansion
  8. Graphical notations
  9. $\text{Self-energies}$
  10. Definition of the $T$-expansion
  11. Definition of the ${\text{$T$-equation}}$
  12. Basic graph operations
  13. Dotted edge operations
  14. Weight expansion
  15. Edge expansions
  16. ...and 26 more sections

Key Result

Theorem 1.3

Fix $d\geqslant 8$, small constants $c_0, c_1, \gamma, \kappa>0$ and a large constant $K > 1$. Suppose that $W\leqslant \ell \leqslant L^{1 - c_0}$, $L^{c_1} \leqslant W \leqslant L$ and $H$ is a $d$-dimensional random band matrix satisfying Assumptions assmH and var profile. Then we have that for provided that $L$ is sufficiently large depending on these constants. Moreover, for any eigenvalue

Figures (3)

  • Figure 1: The flow chart for local expansions. If the weight, multi-edge, $GG$, or $G\overline G$ expansion does not do anything to an input graph (in which case we call it a null operation), then we have "No" and send it to the next operation. In particular, if all graph operations are null for a graph, then it is locally standard and will be sent to the output. On the other hand, if a non-trivial graph operation is acted on an input graph, then we have "Yes" and we will check whether the resulting graphs satisfy the stopping rules (S2)--(S4). If a graph indeed satisfies the stopping rules, then we send it to the output. Otherwise, we send it back to the first step $\mathcal{O}_{dot}$.
  • Figure 2: The main structure of the proof of Theorem \ref{['main thm']}. Corresponding to \ref{['Lcondition0']}, we denote $L_n:=W^{1+(n-2)d/{4} - {c_0}/{2}}$ for a constant $c_0>0$. Moreover, given the $L$ in Theorem \ref{['main thm']}, it is enough to perform the induction up to $n_{W,L}:= \left\lceil\frac{4}{d}\left(\log_W L - 1 + \frac{c_0}{2}\right)\right\rceil+2$.
  • Figure 3: The structure of the proof of Theorem \ref{['thm ptree']}, where $\eta_k$ is defined in \ref{['def etak']}.

Theorems & Definitions (130)

  • Theorem 1.3: Weak delocalization of bulk eigenvectors in high dimesnions
  • Theorem 1.4: Local law
  • Theorem 1.5: Quantum diffusion of the $T$-matrix
  • Corollary 1.6: Quantum diffusion
  • Corollary 1.7
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3: Stochastic domination and high probability event
  • Lemma 2.4: $\Theta$-expansion
  • ...and 120 more