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 $N\to \infty$ through even integers, the largest eigenvalue of the model satisfies $$\frac{\lambda_1}{\sqrt{N}}\to 4\int_0^\infty g_0(t)^4\,\mathrm{d}t \approx 0.32504 \qquad \mbox{almost surely}\,,$$ 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, $\int E^2\rho_0(\mathrm{d}E)=1/4$, and $g_0^3\in L^1(0,\infty)$. 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.