The Edge of Random Tensor Eigenvalues with Deviation

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 compare the edge of the genuine eigenvalue distribution with that of the signed distribution, using field theoretic methods at large N combined with earlier rigorous results of [arXiv:1003.1129]. We support our claims with Monte Carlo simulations. We find two critical points as the variance of the noise increases, 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 trivial eigenvalue and they both become complex.