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.

arXiv commentv2, references added

References in corpus (6)

• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Quantum criticality
• Critical behavior of colored tensor models in the large N limit
• Colored Tensor Models - a Review
• The Number of Eigenvalues of a Tensor
• Product of Ginibre matrices: Fuss-Catalan and Raney distributions

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