Banded non-Hermitian random matrices, neural networks, and eigenvalue degeneracies (opens in new tab)

We study two-banded, non-Hermitian random matrices inspired by sparse neural networks with a circular, 1d topology. We focus on two paradigmatic models, an SSH chain and a ladder model, which have both non-Hermitian directional bias and random sign disorder in the hoppings. The random sign disorder, which follows Dale's Law, leads to localization of the eigenstates, while the directional bias drives a delocalization transition in these states....