# Quartic SYK spectral edge — H6

For each even \(N\ge4\), let \(q_1,\ldots,q_N\) be self-adjoint matrices in an irreducible representation satisfying \(q_iq_j+q_jq_i=2\delta_{ij}I\). Put \(\Psi_S=-q_{i_1}q_{i_2}q_{i_3}q_{i_4}\) for \(S=\{i_1<i_2<i_3<i_4\}\), and let \(J_S\) be independent standard real Gaussians.
Define
\[
H_N=\binom N4^{-1/2}\sum_{|S|=4}J_S\Psi_S.
\]
There exist a probability measure \(\rho_0\) on \([0,\infty)\) and a constant \(\kappa_4>0\) such that, with
\[
g_0(t)=\frac12\int_0^\infty e^{-Et}\,\rho_0(dE),
\]
one has
\[
\int_0^\infty E^2\,\rho_0(dE)=\frac14,\qquad g_0^3\in L^1(0,\infty),
\]
\[
\int_0^\infty\frac{v}{v^2+E^2}\,\rho_0(dE)
=\frac1{v+2\int_0^\infty\sin(vt)g_0(t)^3\,dt}\qquad(v>0),
\]
and
\[
\frac{\lambda_{\max}(H_N)}{\sqrt N}\longrightarrow
\kappa_4=4\int_0^\infty g_0(t)^4\,dt
\quad\text{almost surely as even }N\to\infty.
\]
The Gaussian families at different \(N\) may be coupled arbitrarily on a common probability space.