Edge statistics for random band matrices
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909633596096512 |
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| author | Liu, Dang-Zheng Zou, Guangyi |
| author_facet | Liu, Dang-Zheng Zou, Guangyi |
| contents | We consider Hermitian and symmetric random band matrices on the $d$-dimensional lattice $(\mathbb{Z}/L\mathbb{Z})^d$ with bandwidth $W$, focusing on local eigenvalue statistics at the spectral edge in the limit $W\to\infty$. Our analysis reveals a critical dimension $d_c=6$ and identifies the critical bandwidth scaling as $W_c=L^{(1-d/6)_+}$. In the Hermitian case, we establish the Anderson transition for all dimensions $d<4$, and GUE edge universality when $d\geq 4$ under the condition $W\geq L^{1/3+ε}$ for any $ε>0$. In the symmetric case, we also establish parallel but more subtle transition phenomena after tadpole diagram renormalization. These findings extend Sodin's pioneering work [Ann. Math. 172, 2010], which was limited to the one-dimensional case and did not address the critical phenomena. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_00492 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Edge statistics for random band matrices Liu, Dang-Zheng Zou, Guangyi Probability Mathematical Physics 60B20 We consider Hermitian and symmetric random band matrices on the $d$-dimensional lattice $(\mathbb{Z}/L\mathbb{Z})^d$ with bandwidth $W$, focusing on local eigenvalue statistics at the spectral edge in the limit $W\to\infty$. Our analysis reveals a critical dimension $d_c=6$ and identifies the critical bandwidth scaling as $W_c=L^{(1-d/6)_+}$. In the Hermitian case, we establish the Anderson transition for all dimensions $d<4$, and GUE edge universality when $d\geq 4$ under the condition $W\geq L^{1/3+ε}$ for any $ε>0$. In the symmetric case, we also establish parallel but more subtle transition phenomena after tadpole diagram renormalization. These findings extend Sodin's pioneering work [Ann. Math. 172, 2010], which was limited to the one-dimensional case and did not address the critical phenomena. |
| title | Edge statistics for random band matrices |
| topic | Probability Mathematical Physics 60B20 |
| url | https://arxiv.org/abs/2401.00492 |