Optimal Lower Bound on Eigenvector Overlaps for non-Hermitian Random Matrices

Understand

We consider large non-Hermitian $N\times N$ matrices with an additive independent, identically distributed (i.i.d.) noise for each matrix elements.

• We show that already a small noise of variance $1/N$ completely thermalises the bulk singular vectors, in particular they satisfy the strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence.
• In physics terms, we thus extend the Eigenstate Thermalisation Hypothesis, formulated originally by [Deutsch 1991] and proven for Wigner matrices in [Cipolloni, Erd\H{o}s, Schr\"oder 2020], to arbitrary non-Hermitian matrices with an i.i.d.
• noise.