The Edge of Random Tensor Eigenvalues with Deviation

TL;DR

This paper analyzes the distribution of the largest eigenvalues of symmetric Gaussian random tensors with noise, identifying critical noise levels where eigenvalues emerge and become complex, supported by theoretical and simulation results.

Contribution

It extends previous work by computing eigenvalue distributions for noisy random tensors and characterizes critical noise thresholds affecting eigenvalue behavior.

Findings

Identification of two critical noise variances affecting eigenvalue emergence and merging.

Demonstration of eigenvalue behavior transition from real to complex as noise increases.

Support of theoretical results with Monte Carlo simulations.

Abstract

The largest eigenvalue of random tensors is an important feature of systems involving disorder, equivalent to the ground state energy of glassy systems or to the injective norm of quantum states. For symmetric Gaussian random tensors of order 3 and of size $N$, in the presence of a Gaussian noise, continuing the work [arXiv:2310.14589], we compute the genuine and signed eigenvalue distributions, using field theoretic methods at large $N$ combined with earlier rigorous results of [arXiv:1003.1129]. We characterize the behaviour of the edge of the two distributions as the variance of the noise increases. We find two critical values of the variance, the first of which corresponding to the emergence of an outlier from the main part of the spectrum and the second where this outlier merges with the corresponding largest eigenvalue and they both become complex. We support our claims with Monte Carlo simulations. We believe that our results set the ground for a definition of pseudospectrum of random tensors based on $Z$-eigenvalues.