An independent symmetric-Gaussian norm moment input

Assume the Axiom of Choice. Let $Y$ be a real symmetric $r\times r$ matrix, $r\ge 1$, whose entries on one triangular half including the diagonal are independent $N(0,1)$ variables. Mirrored entries coincide and the diagonal variance is one. With the vertex Hilbert-space operator norm, for every integer $\mathrm{\ell }\ge 2$,

$(\mathbb{E}\mathrm{\parallel }Y{\mathrm{\parallel }}^{2\mathrm{\ell }}{)}^{1\mathrm{/}(2\mathrm{\ell })}\le 2\sqrt{r}+2\sqrt{2\mathrm{\ell }}\mathrm{.}$

Moreover,

$(\mathbb{E}\mathrm{Tr}(Y^{2\mathrm{\ell }}){)}^{1\mathrm{/}(2\mathrm{\ell })}\le r^{1\mathrm{/}(2\mathrm{\ell })}(2\sqrt{r}+2\sqrt{2\mathrm{\ell }})\mathrm{.}$

For this same real symmetric variance-one triangular Gaussian matrix, set $S=\{v\in {\mathbb{R}}^r:\mathrm{\parallel }v{\mathrm{\parallel }}_2=1\}$. With ${\lambda }_{+}={\sup }_{v\in S}{v}^{T}Yv$ and ${\lambda }_{-}=-{\inf }_{v\in S}{v}^{T}Yv$, Bandeira and van Handel prove the following Gaussian-concentration estimate in the proof of Lemma 2.2 (printed/PDF p. 7): for every integer $p\ge 2$, $\mathbb{E}\max (\mathrm{\mid }{\lambda }_{+}-\mathbb{E}{\lambda }_{+}{\mathrm{\mid }}^{2p},\mathrm{\mid }{\lambda }_{-}-\mathbb{E}{\lambda }_{-}{\mathrm{\mid }}^{2p})\le p!4^{p+1}\le (2\sqrt{2p}{)}^{2p}$. This specialized estimate is imported. No local proof of Gaussian concentration is asserted.

The five intermediate items are local proofs, preceding this remaining external import. Their mean comparison follows the method of Lemma 2.2, using the already authored finite-net supplier; the concentration display is explicitly retained as an external premise. Read Lemma 2.2 and its complete two-page proof, checked its variance-one diagonal convention and the following trace use, and inspected the corresponding PDF images. The cited general concentration theorems were not read. No claim of a wholly local concentration proof or independent review is made.