Boundedness of the subleading fourth-moment term below \(\gamma=1/2\)

Establish whether the nonfactorizing fourth-moment contribution \(\Xi_N\) remains bounded as \(N\to\infty\) for the row-independent power-law ensemble when \(0<\gamma<1/2\), using an argument that captures cancellation between covariance terms.

Background

The paper proves $\Xi_N=O(1)$ for $\gamma>1/2$ and proves only that $\Xi_N$ is negligible relative to the leading factored term for every $\gamma>0$.

Below $\gamma=1/2$, the available Cauchy–Schwarz estimate diverges and cannot exploit the sign cancellation that numerical results suggest may keep $\Xi_N$ bounded.