The spectral edge of the quartic SYK model

We consider the Sachdev–Ye–Kitaev model of $N$ Majorana fermions with random $q$-body interactions. For $q=4$, we show that as $N\to \mathrm{\infty }$ through even integers, the largest eigenvalue of the model satisfies $$\frac{{\lambda }_1}{\sqrt{N}}\to 4{\int }_0^{\mathrm{\infty }}{g}_0(t{)}^4\mathrm{d}t\approx 0.32504almostsurely,$$ where $g_0(t)=\frac{1}{2}\int {\mathrm{e}}^{-Et}{\rho }_0(\mathrm{d}E)$ is the unique solution of the zero-temperature quartic Schwinger–Dyson equation for which ${\rho }_0$ is a probability measure, $\int E^2{\rho }_0(\mathrm{d}E)=1\mathrm{/}4$, and $g_0^3\in L^1(0,\mathrm{\infty })$. The main ingredient of the proof is the calculation of the SYK free-energy limit at every fixed positive temperature. We achieve this by introducing a new finite-bath interpolation, which reduces the quartic SYK pressure problem to a local cavity-kernel identity valid at every such temperature. We then identify the zero-temperature slope of the Schwinger–Dyson pressure and transfer it to the spectral edge. GPT-5.6 assisted with literature search, the development of technical arguments, and manuscript preparation; the author is responsible for the contents.