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

TL;DR

This paper proves that small i.i.d. noise causes bulk singular vectors of large non-Hermitian matrices to thermalize rapidly, extending quantum ergodicity results and providing optimal bounds on eigenvector overlaps.

Contribution

It extends the Eigenstate Thermalisation Hypothesis to non-Hermitian matrices with i.i.d. noise and introduces a new decomposition method for analyzing resolvent products.

Findings

Bulk singular vectors satisfy strong QUE with optimal convergence speed

Provides an optimal lower bound on eigenvector overlaps

Develops a new systematic decomposition for resolvent analysis

Abstract

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. 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 [arXiv:2005.08930] and [arXiv:2005.08908]. 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.