Localization-delocalization transition for a random block matrix model at the edge
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913937511940096 |
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| author | Fan, Jiaqi Stone, Bertrand Yang, Fan Yin, Jun |
| author_facet | Fan, Jiaqi Stone, Bertrand Yang, Fan Yin, Jun |
| contents | Consider a random block matrix model consisting of $D$ random systems arranged along a circle, where each system is modeled by an independent $N\times N$ complex Hermitian Wigner matrix. Neighboring systems interact via an arbitrary deterministic $N\times N$ matrix $A$. In this paper, we extend the localization-delocalization transition previously established in arxiv:2312.07297 for the bulk eigenvalue spectrum to the entire spectrum, including the spectral edges. Let $[E^-,E^+]$ denote the support of the limiting spectral density, and define $κ_E:=|E-E^+|\wedge |E-E^-|$ as the distance from a given energy $E \in [E^-, E^+]$ to the spectral edges. We show that for eigenvalues near $E$, the corresponding eigenvectors undergo a localization-delocalization transition when $\|A\|_{\mathrm{HS}}$ crosses the critical threshold $(κ_E + N^{-2/3})^{-1/2}$. In the delocalized phase, the extreme eigenvalues asymptotically follow the Tracy-Widom distribution, while in the localized phase, the edge eigenvalue statistics asymptotically match those of $D$ independent GUE ensembles, up to a deterministic shift. Our results recover those of arxiv:2312.07297 in the bulk regime, where $κ_E \asymp 1$, and further reveal the presence of mobility edges near $E^\pm$ when $1 \ll \|A\|_{\mathrm{HS}} \ll N^{1/3}$. Specifically, bulk eigenvectors corresponding to energies $E$ with $κ_E \gg \|A\|_{\mathrm{HS}}^{-2}$ are delocalized, while those with $κ_E \ll \|A\|_{\mathrm{HS}}^{-2}$ are localized. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00512 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localization-delocalization transition for a random block matrix model at the edge Fan, Jiaqi Stone, Bertrand Yang, Fan Yin, Jun Probability Consider a random block matrix model consisting of $D$ random systems arranged along a circle, where each system is modeled by an independent $N\times N$ complex Hermitian Wigner matrix. Neighboring systems interact via an arbitrary deterministic $N\times N$ matrix $A$. In this paper, we extend the localization-delocalization transition previously established in arxiv:2312.07297 for the bulk eigenvalue spectrum to the entire spectrum, including the spectral edges. Let $[E^-,E^+]$ denote the support of the limiting spectral density, and define $κ_E:=|E-E^+|\wedge |E-E^-|$ as the distance from a given energy $E \in [E^-, E^+]$ to the spectral edges. We show that for eigenvalues near $E$, the corresponding eigenvectors undergo a localization-delocalization transition when $\|A\|_{\mathrm{HS}}$ crosses the critical threshold $(κ_E + N^{-2/3})^{-1/2}$. In the delocalized phase, the extreme eigenvalues asymptotically follow the Tracy-Widom distribution, while in the localized phase, the edge eigenvalue statistics asymptotically match those of $D$ independent GUE ensembles, up to a deterministic shift. Our results recover those of arxiv:2312.07297 in the bulk regime, where $κ_E \asymp 1$, and further reveal the presence of mobility edges near $E^\pm$ when $1 \ll \|A\|_{\mathrm{HS}} \ll N^{1/3}$. Specifically, bulk eigenvectors corresponding to energies $E$ with $κ_E \gg \|A\|_{\mathrm{HS}}^{-2}$ are delocalized, while those with $κ_E \ll \|A\|_{\mathrm{HS}}^{-2}$ are localized. |
| title | Localization-delocalization transition for a random block matrix model at the edge |
| topic | Probability |
| url | https://arxiv.org/abs/2504.00512 |