2.4 Standard analytic tools
Let \(H\) be a polynomial of degree at most \(q\) in either independent Rademacher signs or independent centered Gaussians. Then for every \(p\ge 2\), \(\| H\| _{L^p}\le (p-1)^{q/2}\| H\| _{L^2}\).
For every integer \(m\ge 0\) there is \(C_m{\gt}0\) such that \(\int _{\mathbb {R}^d}\| \lambda \| ^{2m}e^{-2t\| \lambda \| ^2}\, d\lambda \le C_m(d/t)^m F(d,t)\) for all \(d,t\ge 1\).
Let \(A\) be a complex symmetric \(m\times m\) matrix whose real part is positive definite. Then \(\int _{\mathbb {R}^m}e^{-x^\top A x}\, dx=\pi ^{m/2}\det (A)^{-1/2}\). In particular, if \(g\sim N(0,I_m)\) and \(M\) is real symmetric, \(|\mathbb {E}[e^{ig^\top Mg/2}]|=\det (I+M^2)^{-1/4}\).
Diagonalize \(S=\operatorname {Re}(A)^{-1/2}\, \mathrm{Im}(A)\, \operatorname {Re}(A)^{-1/2}\), factor the integral into one-dimensional Fresnel integrals, and reassemble via \(\det (A)=\det (\operatorname {Re}A)\det (I+iS)\).
There exists an absolute constant \(c_*{\gt}0\) such that \(G_{\mathrm{core}}(d,t)\ge (1-e^{-c_*d})F(d,t)\).
On \(\| \lambda \| ^2{\gt}d/t\), one has \(e^{-2t\| \lambda \| ^2}\le e^{-d}e^{-t\| \lambda \| ^2}\), so the tail is at most \(e^{-d}(\pi /t)^{d/2}\le e^{-c_*d}F(d,t)\) with \(c_*=1-\frac12\log 2\).