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

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.

v2, references added

References in corpus (1)

• Quantum criticality

Cited by in corpus (6)

• Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap
• Exact analytic expressions of real tensor eigenvalue distributions of Gaussian tensor model for small $N$
• Spectral Form Factor for Time-dependent Matrix model
• Tensor Field Theories: Renormalization and Random Geometry
• Functional Renormalization Group Approach for Signal Detection
• Melonic large $N$ limit of $5$-index irreducible random tensors