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

Giorgio Cipolloni, László Erdős, Joscha Henheik, Dominik Schröder

Preprint(2023)

## Summary

We show that already a small noise completely thermalizes the bulk singular vectors of arbitrary deterministic matrices. In particular we prove a strong form of Quantum Unique Ergodicity (QUE) with an optimal speed of convergence for such matrices.## Abstract

We consider large non-Hermitian $N × N$ 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 thermalizes 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 [34] and proven for Wigner matrices in [24], to arbitrary non-Hermitian matrices with an i.i.d. noise. As a consequence we obtain an optimal lower bound on the diagonal overlaps of the corresponding non-Hermitian eigenvectors. This quantity, also known as the (square of the) eigenvalue condition number measuring the sensitivity of the eigenvalue to small perturbations, has notoriously escaped rigorous treatment beyond the explicitly computable Ginibre ensemble apart from the very recent upper bounds given in [7] and [45]. As a key tool, we develop a new systematic decomposition of general observables in random matrix theory that governs the size of products of resolvents with deterministic matrices in between.