Three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors
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| Format: | Preprint |
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2024
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| author | Majumder, Swastik Sasakura, Naoki |
| author_facet | Majumder, Swastik Sasakura, Naoki |
| contents | Random tensor models have applications in a variety of fields, such as quantum gravity, quantum information theory, mathematics of modern technologies, etc., and studying their statistical properties, e.g., tensor eigenvalue/vector distributions, are interesting and useful. Recently some tensor eigenvalue/vector distributions have been computed by expressing them as partition functions of zero-dimensional quantum field theories. In this paper, using the method, we compute three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors, where the three cases can be characterized by the Lie-group invariances, $O(N,\mathbb{R})$, $O(N,\mathbb{C})$, and $U(N,\mathbb{C})$, respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the "signed" distribution which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-$N$ limit by computing the edge of the last signed distribution, obtaining agreement with a former numerical result in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_01030 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors Majumder, Swastik Sasakura, Naoki High Energy Physics - Theory Mathematical Physics Random tensor models have applications in a variety of fields, such as quantum gravity, quantum information theory, mathematics of modern technologies, etc., and studying their statistical properties, e.g., tensor eigenvalue/vector distributions, are interesting and useful. Recently some tensor eigenvalue/vector distributions have been computed by expressing them as partition functions of zero-dimensional quantum field theories. In this paper, using the method, we compute three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors, where the three cases can be characterized by the Lie-group invariances, $O(N,\mathbb{R})$, $O(N,\mathbb{C})$, and $U(N,\mathbb{C})$, respectively. Exact closed-form expressions of the distributions are obtained by computing partition functions of four-fermi theories, where the last case is of the "signed" distribution which counts the distribution with a sign factor coming from a Hessian matrix. As an application, we compute the injective norm of the complex symmetric order-three random tensor in the large-$N$ limit by computing the edge of the last signed distribution, obtaining agreement with a former numerical result in the literature. |
| title | Three cases of complex eigenvalue/vector distributions of symmetric order-three random tensors |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2408.01030 |