Finite-dimensional fluctuations of overlap and Rayleigh quotient

Determine the dimension-dependent variance, and more generally characterize the finite-dimensional fluctuations, of the squared overlap and Rayleigh quotient produced by low-degree polynomial approximations to the top eigenvector and eigenvalue of GOE and spiked GOE matrices.

Background

The paper’s main theorems characterize expected squared overlap and expected Rayleigh quotient in asymptotic regimes where the matrix dimension and polynomial degree grow. Numerical experiments, however, show substantial trial-to-trial variation even at dimensions such as 16000. The unresolved problem is to quantify this variability as a function of dimension and polynomial degree, rather than only determining its limiting expectation.

References

This motivates the question of understanding the size of the variance of squared overlap / Rayleigh quotient as dimension-dependent random variables. We leave this for future work.

— Eigenvalue and Eigenvector Approximation for Random Matrices Using Low-Degree Polynomials  (2609.28781 - Zhang, 23 Sep 2026) in Section 7, Discussion and future directions, paragraph “Fluctuation”