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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2510.04068 |
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| _version_ | 1866914075922923520 |
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| author | Delporte, Nicolas La Scala, Giacomo Sasakura, Naoki Toriumi, Reiko |
| author_facet | Delporte, Nicolas La Scala, Giacomo Sasakura, Naoki Toriumi, Reiko |
| contents | We propose a new definition of characteristic polynomials of tensors based on a partition function of Grassmann variables. This new notion of characteristic polynomial addresses general tensors including totally antisymmetric ones, but not totally symmetric ones. Drawing an analogy with matrix eigenvalues obtained from the roots of their characteristic polynomials, we study the roots of our tensor characteristic polynomial. Unlike standard definitions of eigenvalues of tensors of dimension $N$ giving $\sim e^{\text{constant} \, N}$ number of eigenvalues, our polynomial always has $N$ roots. For random Gaussian tensors, the density of roots follows a generalized Wigner semi-circle law based on the Fuss-Catalan distribution, introduced previously by Gurau [arXiv:2004.02660 [math-ph]]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04068 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characteristic polynomials of tensors via Grassmann integrals and distributions of roots for random Gaussian tensors Delporte, Nicolas La Scala, Giacomo Sasakura, Naoki Toriumi, Reiko Mathematical Physics High Energy Physics - Theory We propose a new definition of characteristic polynomials of tensors based on a partition function of Grassmann variables. This new notion of characteristic polynomial addresses general tensors including totally antisymmetric ones, but not totally symmetric ones. Drawing an analogy with matrix eigenvalues obtained from the roots of their characteristic polynomials, we study the roots of our tensor characteristic polynomial. Unlike standard definitions of eigenvalues of tensors of dimension $N$ giving $\sim e^{\text{constant} \, N}$ number of eigenvalues, our polynomial always has $N$ roots. For random Gaussian tensors, the density of roots follows a generalized Wigner semi-circle law based on the Fuss-Catalan distribution, introduced previously by Gurau [arXiv:2004.02660 [math-ph]]. |
| title | Characteristic polynomials of tensors via Grassmann integrals and distributions of roots for random Gaussian tensors |
| topic | Mathematical Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2510.04068 |