Eigenvalue instantons in the spectral form factor of random matrix model
Kazumi Okuyama
TL;DR
The paper investigates the late-time plateau of the spectral form factor (SFF) in the Gaussian unitary ensemble (GUE) and demonstrates that non-perturbative corrections in the $1/N$ expansion arise from eigenvalue instantons that move eigenvalues from the Wigner semicircle cut to imaginary saddle points. It derives the instanton action $S_{\text{inst}}(\tau)=2[\tau\sqrt{\tau^2-1}-\operatorname{arccosh}(\tau)]$ and shows the connected part of the SFF gains corrections of order $e^{-2N S_{\text{inst}}(\tau)}$, with a two-instanton saddle dominating in the plateau. A differential-equation approach for the Laguerre-based one-point function yields a controlled $1/N$ expansion around the instanton, giving explicit forms for $\mathcal{Z}(\tau)$ and subsequent corrections to $\partial_\tau g_{\text{conn}}(\tau)$. The results illuminate a non-perturbative mechanism for ramp-plateau transitions, connect the problem to the Douglas-Kazakov transition in 2d Yang-Mills on $S^2$ and to large-winding Wilson loops in $\mathcal{N}=4$ SYM, and outline directions for extending to other ensembles and to SYK holography.
Abstract
We study the late time plateau behavior of the spectral form factor in the Gaussian Unitary Ensemble (GUE) random matrix model. The time derivative of the spectral form factor in the plateau regime is not strictly zero, but non-zero due to a non-perturbative correction in the $1/N$ expansion. We argue that such a non-perturbative correction comes from the eigenvalue instanton of random matrix model and we explicitly compute the instanton correction as a function of time.