Local law and delocalization for the Sachdev-Ye-Kitaev model

Abstract

We establish a mesoscopic local law and quantitative eigenvector-delocalization estimates for the Sachdev--Ye--Kitaev model (SYK) Hamiltonian of $N$ interacting Majorana fermions. For even $q\ll N^{\frac{1}{2}}$, we prove that the normalized Stieltjes transform converges uniformly on bounded energy intervals to that of the standard Gaussian law, down to scales of order $qN^{-\frac{1}{2}}$, up to logarithmic factors. The result holds both on the full Hilbert space and in each fermion-parity sector. We derive mesoscopic eigenvalue counting and spectral form factor estimates, as well as an averaged inverse participation ratio bound. For fixed $q$, we further obtain high-probability $\ell^\infty$-delocalization bounds in any deterministic orthonormal basis for individual bulk eigenvectors. These are the first mesoscopic laws and eigenvector delocalization estimates for the SYK Hamiltonian, which we obtain using the resolvent method.