The spectral edge of the quartic SYK model

Abstract

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 \[ \frac{λ_1}{\sqrt{N}}\to 4\int_0^\infty g_0(t)^4\,\mathrm{d}t \approx 0.32504 \qquad \mbox{almost surely}\,, \] where $g_0(t)=\frac{1}{2}\int \mathrm{e}^{-Et}ρ_0(\mathrm{d}E)$ is the unique solution of the zero-temperature quartic Schwinger--Dyson equation for which $ρ_0$ is a probability measure, $\int E^2ρ_0(\mathrm{d}E)=1/4$, and $g_0^3\in L^1(0,\infty)$. The main technical result of the proof is the calculation of the SYK free-energy limit at all positive temperatures. The proof contains three new ingredients: a single-site cavity expansion that keeps the bulk Gibbs state intact, a finite-dimensional locality estimate yielding label-uniform conditional factorization of the Euclidean cavity fields, and an exact quadratic Majorana-bath representation of the leading diagrams. Anti-monotonicity of the Dyson map against strict monotonicity of the cube forces the limiting kernel to be the Schwinger--Dyson kernel, which allows us to compute the free energy with no temperature threshold. Our method also applies to other fixed even $q\geq 6$. GPT-5.6 assisted with literature search, the development of technical arguments, and manuscript preparation; the author is responsible for the contents.