3 Weak comparison
The invariance-principle input: a Lindeberg replacement argument that compares the Rademacher characteristic function \(\psi \) with its Gaussian counterpart \(\psi _G\), with an error controlled by row influences.
Let \(f:[n]^2\to \mathbb {R}\) be symmetric with \(f(i,i)=0\), and define
If \(X_1,\dots ,X_n\) are independent Rademacher signs, \(G_1,\dots ,G_n\) are i.i.d. \(N(0,1)\), and \(\varphi :\mathbb {R}\to \mathbb {R}\) is \(C^3\), then
Write \(Z^{(m)}:=(G_1,\dots ,G_m,X_{m+1},\dots ,X_n)\) and telescope \(\mathbb {E}[\varphi (Q_f(X))]-\mathbb {E}[\varphi (Q_f(G))] =\sum _{m=1}^n(F_{m-1}-F_m)\). Decompose \(Q_f(z)=U_m(z)+z_mV_m(z)\) where \(V_m\) depends only on coordinates other than \(m\). Taylor-expand to third order; the zeroth and second moments of \(X_m\) and \(G_m\) match, so the first two Taylor terms cancel. The remainder is bounded using \(\mathbb {E}[|V_m|^3]\le 8\sqrt3\, \mathrm{Inf}_m(f)^{3/2}\) (Cauchy–Schwarz on the second and fourth moments of \(V_m\)). Summing over \(m\) gives the claim.
Let \(\psi _G\) be the characteristic function obtained by replacing the Rademacher entries \(\xi _i\) with i.i.d. standard Gaussians \(g_i\). For each row \(k\in [n]\), define the row influence \(I_k(\lambda ):=\sum _{i\ne k}\lambda _{\{ i,k\} }^2\), and set \(J(\lambda ):=\sum _{k=1}^n I_k(\lambda )^{3/2}\). Then
Set \(f(i,j)=\lambda _{\{ i,j\} }/2\) so that \(\mathrm{Inf}_m(f)=\tfrac 14 I_m(\lambda )\). Apply Lemma 12 with \(\varphi =\cos \) and \(\varphi =\sin \), using \(\| \cos '''\| _\infty ,\| \sin '''\| _\infty \le 1\), then combine real and imaginary parts.