Understanding an argument involving the Sherrington-Kirkpatrick spin glass model

Question

I have a question regarding an argument in

Aizenman, M.; Lebowitz, J. L.; Ruelle, D., Some rigorous results on the Sherrington-Kirkpatrick spin glass model., Commun. Math. Phys. 112, No. 1, 3-20 (1987); addendum 116, No. 3, 527 (1988). ZBL1108.82312.

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In this paper the following random function is studied

\begin{equation} H(\sigma)= \dfrac{1}{\sqrt{n}}\sum \limits_{1 \leq i < j \leq N} J_{ij}\sigma_i \sigma_j \end{equation}

where $\sigma_1, \ldots, \sigma_n$ take values $\pm 1$ and the $J_{ij}$'s are independent identically distributed random variables with mean zero and variance $J^{2}$.

I'm interested in the following argument on page 19:

where the bound stated from equation (5.1) is

and the notation $\langle \langle \ \rangle \rangle$ stands for expectation.

Question:

I can follow all the arguments but I am stuck at understanding why we can apply a bound like $(5.1)$ to $H_J(\sigma) - \tilde{\lambda} \langle \tilde{\phi},\sigma \rangle^{2}$. In particular I would be interested in a detailed explanation of why/how the invariance of the distribution of the frame of eigenvectors under general unitary rotations implies this result.

Answer 1

I would argue as follows: unitary invariance means you are free to choose a basis. Let me choose the basis where $\tilde\phi=\{1,0,0,\ldots,0,0\}$, then $H_J(\sigma)-\tilde\lambda\langle\tilde\phi|\sigma\rangle^2=H_J(\sigma)+J$, and I can then apply equation (5.1) to $$\langle\langle(H_J(\sigma)+J)^2\rangle\rangle=\langle\langle H_J(\sigma)^2\rangle\rangle+J^2=O(N),$$ because $\langle\langle H_J(\sigma)\rangle\rangle=0$.