Spectral edge of the quartic SYK model

The problem asks for the leading growth of the largest eigenvalue in the quartic SYK Hamiltonian. A new preprint gives an almost-sure limit as $N$ tends to infinity through even integers.

Known results

• The solvable quadratic case has $4\sqrt{2N}/(3\pi)+o(\sqrt{N})\u00024$ for the largest eigenvalue.
• For fixed even interaction order at least $4$, earlier work had only the rough bound $\mathbb{E}\lambda_{\max}\leq\sqrt{N\log 2}\u0002$ and no limiting edge constant.

July 2026 resolution

The paper “The spectral edge of the quartic SYK model” proves

$$\frac{\lambda_1}{\sqrt{N}}\longrightarrow\kappa_{\rm SD}$$

almost surely, where $\kappa_{\rm SD}=4\int_0^\infty g_0(t)^4\,\mathrm{d}t\approx0.3250421580667593\u0002$. Thus $\lambda_1\sim\kappa_{\rm SD}\sqrt{N}\u0002$. The proof derives the fixed-temperature free-energy limit and transfers its zero-temperature slope to the spectral edge.

Current status (as of July 2026): the quartic spectral edge is settled almost surely, with $\lambda_1\sim0.3250421580667593\sqrt{N}\u0002$; no aspect of the stated problem remains open.