Thermodynamic limit "vs" the method of steepest descent Let me use this lecture note as the reference.

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*I would like to know how in the above the expression (14) was obtained from expression (12).

In some sense it makes intuitive sense but I would like to know of the details of what happened in between those two equations. The point being that if there were no overall factor of "$\sqrt{N}$" in equation (12) then it would be a "textbook" case of doing the "method of steepest descent" in the asymptotic limit of "N".

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*I am wondering if in between there is an unwritten argument that in the "large N" limit one is absorbing the $\sqrt{N}$ into a re(un?)defined measure and then doing the steepest descent on only the exponential part of the integrand.
I don't know how is the "method of steepest descent" supposed to be done on the entire integrand if the measure were not to be redefined.


*But again if something like that is being done then why is there an approximation symbol in equation (14)?
After taking the thermodynamic limit and doing the steepest descent shouldn't the equation (14) become an equality with a sum over all the $\mu_s$ which solve equation (15)?
Though naively to me the expression (12) looks more amenable to a Dirac delta function interpretation since in the "large N" limit it seems to be looking like the standard representation of the Dirac delta function, $\frac{n}{\sqrt{\pi}} e^{-n^2x}$

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*I would like to know of some comments/explanation about this general philosophy/proof by which one wants to say that the "method of steepest descent" is exact in the "thermodynamic limit".

 A: This is a straightforward application of the steepest descent method. So, we have the integral (taken from the lectures you cite)
$$Q_N=\sqrt{\frac{N\beta J}{2\pi}}\int_{-\infty}^\infty d\mu e^{[Nq(\beta J,\beta b,\mu)]}$$
being
$$q(\beta J,\beta b,\mu)=\ln\{2\cosh[\beta(J\mu+b)]\}-\frac{\beta J\mu^2}{2}.$$
Now, you will have, by applying steepest descent method to the integral,
$$\frac{\partial q}{\partial\mu}=\beta J\tanh[\beta(J\mu+b)]-\beta J\mu=0.$$
Let us call $\mu_s$ the solution of this equation and expand the argument of the exponential around this value. You will get
$$q(\beta J,\beta b,\mu)=q(\beta J,\beta b,\mu_s)-\frac{J\beta}{2}[1-J\beta(1-\mu_s^2)](\mu-\mu_s)^2.$$ 
This shows that
$$Q_N\approx e^{Nq(\beta J,\beta b,\mu_s)}\sqrt{\frac{N\beta J}{2\pi}}\int_{-\infty}^\infty d\mu e^{-\frac{NJ\beta}{2}[1-J\beta(1-\mu_s^2)](\mu-\mu_s)^2}$$
i.e.
$$Q_N\approx e^{Nq(\beta J,\beta b,\mu_s)}\sqrt{\frac{1}{1-J\beta(1-\mu_s^2)}}$$
Note that the $N$ factor is completely removed after integration and remains just into the argument of the exponential.  Eq. (16-18) follow straightforwardly.
