On page 2 of this preprint, the authors make a derivation "by saddle point".
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$Z \propto \prod_{\sigma=1}^{q} dx_\sigma e^{-N \left[\sum_{\sigma} \frac{\beta Jx^2_{\sigma}}{2} - \log\left(\sum_{\sigma} e^{\beta J x_\sigma}\right) \right] }$ (3)
$\text{From Eq. (3), for } N \rightarrow \infty \text{, by saddle point, we get immediately the following system of equations}$:
$x_{\sigma} = \frac{e^{\beta J x_{\sigma}}}{\sum_{\sigma'} e^{\beta J x_{\sigma'}}} \text{, } \sigma = 1, \ldots, q $ (4)
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$$Z \propto \prod_{\sigma=1}^{q} dx_\sigma \, \exp{\left(-N \left[\sum_{\sigma} \frac{\beta Jx^2_{\sigma}}{2} - \log\left(\sum_{\sigma} e^{\beta J x_\sigma}\right) \right] \right) } \tag{3} $$
From Eq. (3), for $N \rightarrow \infty$ by saddle point, we get immediately the following system of equations:
$$x_{\sigma} = \frac{e^{\beta J x_{\sigma}}}{\sum_{\sigma'} e^{\beta J x_{\sigma'}}} \text{, } \sigma = 1, \ldots, q \tag{4}$$
Z$Z$ is the partition function for a q-state mean-field Potts model.
There There are q$q$ independent Gaussian variables x$x$.
Can someone unpack how How does (4) followsfollow from (3)? ideally expanding with some in-between steps.
Many thanks in advance!
