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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Online-Zugang: | https://arxiv.org/abs/2312.07297 |
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| _version_ | 1866913694449926144 |
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| author | Stone, Bertrand Yang, Fan Yin, Jun |
| author_facet | Stone, Bertrand Yang, Fan Yin, Jun |
| contents | Consider $D$ random systems that are modeled by independent $N\times N$ complex Hermitian Wigner matrices. Suppose they are lying on a circle and the neighboring systems interact with each other through a deterministic matrix $A$. We prove that in the asymptotic limit $N\to \infty$, the whole system exhibits a quantum chaos transition when the interaction strength $\|A\|_{HS}$ varies. Specifically, when $\|A\|_{HS}\ge N^{\varepsilon}$, we prove that the bulk eigenvalue statistics match those of a $DN\times DN$ GUE asymptotically and each bulk eigenvector is approximately equally distributed among the $D$ subsystems with probability $1-o(1)$. These phenomena indicate quantum chaos of the whole system. In contrast, when $\|A\|_{HS}\le N^{-\varepsilon}$, we show that the system is integrable: the bulk eigenvalue statistics behave like $D$ independent copies of GUE statistics asymptotically and each bulk eigenvector is localized on only one subsystem. In particular, if we take $D\to \infty$ after the $N\to \infty$ limit, the bulk statistics converge to a Poisson point process under the $DN$ scaling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_07297 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A random matrix model towards the quantum chaos transition conjecture Stone, Bertrand Yang, Fan Yin, Jun Probability Mathematical Physics Consider $D$ random systems that are modeled by independent $N\times N$ complex Hermitian Wigner matrices. Suppose they are lying on a circle and the neighboring systems interact with each other through a deterministic matrix $A$. We prove that in the asymptotic limit $N\to \infty$, the whole system exhibits a quantum chaos transition when the interaction strength $\|A\|_{HS}$ varies. Specifically, when $\|A\|_{HS}\ge N^{\varepsilon}$, we prove that the bulk eigenvalue statistics match those of a $DN\times DN$ GUE asymptotically and each bulk eigenvector is approximately equally distributed among the $D$ subsystems with probability $1-o(1)$. These phenomena indicate quantum chaos of the whole system. In contrast, when $\|A\|_{HS}\le N^{-\varepsilon}$, we show that the system is integrable: the bulk eigenvalue statistics behave like $D$ independent copies of GUE statistics asymptotically and each bulk eigenvector is localized on only one subsystem. In particular, if we take $D\to \infty$ after the $N\to \infty$ limit, the bulk statistics converge to a Poisson point process under the $DN$ scaling. |
| title | A random matrix model towards the quantum chaos transition conjecture |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2312.07297 |