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Recently, I study replica method derived from statistical physics. I have a confusion on following equation \begin{align} \mathcal{F}=-\lim_{M\rightarrow \infty}\frac{1}{M}\lim_{\tau\rightarrow 0}\frac{\partial }{\partial \tau} \underset{\mathbf{Q},\tilde{\mathbf{Q}}}{\sup\inf}\left\{\alpha^{-1}G(\mathbf{Q})-\text{tr}\left\{\mathbf{Q}\tilde{\mathbf{Q}}\right\}-\frac{1}{M}\log \mathbb{E}_{\mathbf{X}}\left\{\exp \left[\text{tr}(\mathbf{X}\tilde{\mathbf{Q}}\mathbf{X}^T)\right]\right\}\right\} \end{align} where $\mathbf{Q}=(c-q)\mathbf{I}_{\tau+1}+q\boldsymbol{ee}^T$ and $\tilde{\mathbf{Q}}=(\tilde{c}-\tilde{q})\mathbf{I}_{\tau+1}+\tilde{q}\boldsymbol{ee}^T$ with $\boldsymbol{e}\in \mathbb{R}^{\tau+1}$ denoting a column vector whose elements are all 1.

In the most of paper about replica method, they always change the sup, inf and partial derivative like following \begin{align} \mathcal{F}=-\lim_{M\rightarrow \infty}\frac{1}{M} \underset{\mathbf{Q},\tilde{\mathbf{Q}}}{\sup\inf}\left\{ \lim_{\tau\rightarrow 0}\frac{\partial }{\partial \tau}\left[ \alpha^{-1}G(\mathbf{Q})-\text{tr}\left\{\mathbf{Q}\tilde{\mathbf{Q}}\right\}-\frac{1}{M}\log \mathbb{E}_{\mathbf{X}}\left\{\exp \left[\text{tr}(\mathbf{X}\tilde{\mathbf{Q}}\mathbf{X}^T)\right]\right\} \right] \right\} \end{align}

What I have thought?

As far as I know, I consider that both $\sup$ and $\inf$ are one limit. Generally, $\lim$ and partial derivative can be interchanged.

I hope you can help me. This is extremely important for me. Thank you!

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