Lemma 11.28.28D640C26806F

Proof. In a $g$-unitary frame, let $Z_1,\ldots ,Z_ n$ be independent standard complex Gaussian variables, with density $\pi ^{-1}\mathrm e^{-|z|^2}$ on $\mathbb C$. Their second and fourth moments are $\mathbb E|Z_ i|^2=1$ and $\mathbb E|Z_ i|^4=2$. Independence and invariance under coordinate phases give

Thus $\mathbb E R^ h(Z,\bar Z,Z,\bar Z)=Q_ g(h)+B_ g(h)$ and $\mathbb E|Z|_ h^4=u^2+q$. Taking expectations in the homogeneous inequality $R^ h(Z,\bar Z,Z,\bar Z)\leq -\kappa |Z|_ h^4$ proves the result. $\square$