Consider the following integral:

\begin{equation}
\int \mathrm{d}\rho \frac{1}{\rho} e^{N f(\rho)}
\end{equation}
Where:
\begin{equation}
f(\rho)=\ln \rho-\frac{1}{2} \rho^{2}+\frac{1}{2 p w^{2}} \rho^{2 p}\implies f^{\prime}(\rho)=\frac{1-\rho^{2}+\frac{1}{w^{2}} \rho^{2 p}}{\rho}
\end{equation}

To compute this integral in the limit for large $N$ I can use the saddle point method which consists in finding a $\rho_0$ such that $f^{\prime}(\rho_0)=0$.

In the following paper [1] they explain that this equation admits a unique physical solution (which remains at finite distance from the origin when sending $w \to\infty$. There is a critical point $w_c$ and the select the root which behaves like $w^{-1}$ at large $w$. They find that 
\begin{equation}
w_c^2=p^{p} /(p-1)^{p-1}
\end{equation}

How can I recover this result? I tried something like this:

\begin{align}
f’(\rho)=0\implies w^2&=\rho^{2 p}/(-1 + \rho^2)\\
\frac{1}{w^2}=\frac{\rho^2-1}{\rho^{2p}}
\end{align}

Sending $w\to \infty$ would be similar to some Taylor expansion I suppose but I am unable to see how to proceed next.


[1] <https://arxiv.org/pdf/2004.02660.pdf>