Sharp Bounds on Ground State Energy of the SYK Model

We study the Sachdev-Ye-Kitaev (SYK) Hamiltonian $H_{\operatorname{SYK}}$ on $n$ Majorana modes with $k$-body interactions, and prove that $\mathbb{E}\|H_{\operatorname{SYK}}\|_{\operatorname{op}} = (1 - o(1))\cdot\sqrt{2n}/k$ for super-constant $k\leq o(\sqrt{n})$, where the expectation is over the