Weak local law and delocalization for the Sachdev-Ye-Kitaev model

Abstract

We establish a weak mesoscopic local law and quantitative eigenvector-delocalization estimates for the SYK Hamiltonian. 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 $q{N}^{-\frac{1}{2}}$, 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.

26 pages