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Delocalization of Two-Dimensional Random Band Matrices

Sofiia Dubova, Kevin Yang, Horng-Tzer Yau, Jun Yin

Abstract

We study a random band matrix $H=(H_{xy})_{x,y}$ of dimension $N\times N$ with mean-zero complex Gaussian entries, where $x,y$ belong to the discrete torus $(\mathbb{Z}/\sqrt{N}\mathbb{Z})^{2}$. The variance profile $\mathbb{E}|H_{xy}|^{2}=S_{xy}$ vanishes when the distance between $x,y$ is larger than some band-width parameter $W$ depending on $N$. We show that if the band-width satisfies $W\geq N^{\mathfrak{c}}$ for some $\mathfrak{c}>0$, then in the large-$N$ limit, we have the following results. The first result is a local semicircle law in the bulk down to scales $N^{-1+\varepsilon}$. The second is delocalization of bulk eigenvectors. The third is a quantum unique ergodicity for bulk eigenvectors. The fourth is universality of local bulk eigenvalue statistics. The fifth is a quantum diffusion profile for the associated $T$ matrix. Our method is based on embedding $H$ inside a matrix Brownian motion $H_{t}$ as done in [Dubova-Yang '24] and [Yau-Yin '25] for band matrices on the one-dimensional torus. In this paper, the key additional ingredient in our analysis of $H_{t}$ is a new CLT-type estimate for polynomials in the entries of the resolvent of $H_{t}$.

Delocalization of Two-Dimensional Random Band Matrices

Abstract

We study a random band matrix of dimension with mean-zero complex Gaussian entries, where belong to the discrete torus . The variance profile vanishes when the distance between is larger than some band-width parameter depending on . We show that if the band-width satisfies for some , then in the large- limit, we have the following results. The first result is a local semicircle law in the bulk down to scales . The second is delocalization of bulk eigenvectors. The third is a quantum unique ergodicity for bulk eigenvectors. The fourth is universality of local bulk eigenvalue statistics. The fifth is a quantum diffusion profile for the associated matrix. Our method is based on embedding inside a matrix Brownian motion as done in [Dubova-Yang '24] and [Yau-Yin '25] for band matrices on the one-dimensional torus. In this paper, the key additional ingredient in our analysis of is a new CLT-type estimate for polynomials in the entries of the resolvent of .

Paper Structure

This paper contains 27 sections, 35 theorems, 428 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The model and main results
  3. Band matrix model
  4. Main results
  5. Universality
  6. Stochastic flow and the loop hierarchy
  7. The propagator Theta and an estimate on K
  8. Estimates on loops
  9. Strategy of the proofs of main lemmas
  10. Notations
  11. Basic properties of K
  12. G-chains and their estimates
  13. Analysis of loop hierarchy
  14. Proof of Theorem \ref{['lem:main_ind']}: Step 1
  15. Case 1: $s < t \leq 1/2$.
  16. ...and 12 more sections

Key Result

Theorem 2.2

Suppose that the following is satisfied for some fixed constant $\mathfrak c>0$: Then for any $\kappa, \tau, D>0$, there exists $N_0$ such that for all $N \geqslant N_0$ we have

Figures (4)

  • Figure 1: Illustration of ${\cal G}^{(a)}_{k}$ for $n=4$. Taken from YY_25.
  • Figure 2: Illustration of operator ${\cal G}^{(a), L}_{k, l}$. Taken from YY_25.
  • Figure 3: Illustration of operator ${\cal G}^{(a), R}_{k, l}$. Taken from YY_25.
  • Figure 4: Example of the $8-G$ loop in $\mathcal{E}\otimes \mathcal{E}$: for $\sigma\in \{+,-\}^3$. Taken from YY_25.

Theorems & Definitions (64)

  • Definition 2.1
  • Theorem 2.2: Delocalization
  • Theorem 2.3: Local semicircle law
  • Theorem 2.4: Generalized quantum unique ergodicity
  • Theorem 2.5: Quantum diffusion
  • proof : Proof of Theorem \ref{['MR:QUE']}
  • Theorem 2.6: Bulk universality
  • proof : Proof of Theorem \ref{['Thm: B_Univ']}
  • Definition 2.7
  • Lemma 2.8
  • ...and 54 more