On the generalization of the Wigner semicircle law to real symmetric tensors
We propose a simple generalization of the matrix resolvent to a resolvent for real symmetric tensors $T\in \otimes^p \mathbb{R}^N$ of order $p\ge 3$. The tensor resolvent yields an integral representation for a class of tensor invariants and its singular locus can be understood in terms of the real...
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| Main Author: | |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
14 Apr 2020
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| Edition: | Version v2 |
| In: |
Arxiv
Year: 2020, Pages: 1-39 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2004.02660
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| Author Notes: | Razvan Gurau |
| Summary: | We propose a simple generalization of the matrix resolvent to a resolvent for real symmetric tensors $T\in \otimes^p \mathbb{R}^N$ of order $p\ge 3$. The tensor resolvent yields an integral representation for a class of tensor invariants and its singular locus can be understood in terms of the real eigenvalues of tensors. We then consider a random Gaussian (real symmetric) tensor. We show that in the large $N$ limit the expected resolvent has a finite cut in the complex plane and that the associated "spectral density", that is the discontinuity at the cut, obeys a universal law which generalizes the Wigner semicircle law to arbitrary order. Finally, we consider a spiked tensor for $p\ge 3$, that is the sum of a fixed tensor $b\,v^{\otimes p}$ with $v\in \mathbb{R}^N$ (the signal) and a random Gaussian tensor $T$ (the noise). We show that in the large $N$ limit the expected resolvent undergoes a sharp transition at some threshold value of the signal to noise ratio $b$ which we compute analytically. |
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| Item Description: | Version 1 vom 6. April 2020, Version 2 vom 14. April 2020 Gesehen am 07.10.2022 |
| Physical Description: | Online Resource |