On the generalization of the Wigner semicircle law to real symmetric tensors

• Verfasst von: Gurǎu, Rǎzvan [VerfasserIn]
• Titel: On the generalization of the Wigner semicircle law to real symmetric tensors
• Verf.angabe: Razvan Gurau
• Ausgabe: Version v2
• E-Jahr: 2020
• Jahr: 14 Apr 2020
• Umfang: 39 S.
• Fussnoten: Version 1 vom 6. April 2020, Version 2 vom 14. April 2020 ; Gesehen am 07.10.2022
• Titel Quelle: Enthalten in: De.arxiv.org
• Ort Quelle: [S.l.] : Arxiv.org, 1991
• Jahr Quelle: 2020
• Band/Heft Quelle: (2020), Artikel-ID 2004.02660, Seite 1-39
• Datenträger: Online-Ressource
• Sprache: eng
• Sach-SW: 60B99
• Sach-SW: High Energy Physics - Theory
• Sach-SW: Mathematical Physics
• K10plus-PPN: 1804434590

Abstract

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.