Delocalization of Two-Dimensional Random Band Matrices
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866929751677992960 |
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| author | Dubova, Sofiia Yang, Kevin Yau, Horng-Tzer Yin, Jun |
| author_facet | Dubova, Sofiia Yang, Kevin Yau, Horng-Tzer Yin, Jun |
| contents | 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 $H_{t}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07606 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Delocalization of Two-Dimensional Random Band Matrices Dubova, Sofiia Yang, Kevin Yau, Horng-Tzer Yin, Jun Probability Mathematical Physics 15B52, 82B44 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 $H_{t}$. |
| title | Delocalization of Two-Dimensional Random Band Matrices |
| topic | Probability Mathematical Physics 15B52, 82B44 |
| url | https://arxiv.org/abs/2503.07606 |