Critical point of saddle point equation

Question

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

Answer 1

You seek a solution $\rho$ of the equation $f'(\rho)=0$, hence $$\rho^2=1+w^{-2}\rho^{2p}.$$ The solution should remain $>0$ when $w\rightarrow\infty$.
_{The OP says the solution should vanish as $1/w$, but that is mistaken, I think.}

To gain some insight, take $p=2$, then the solution is $$\rho=\frac{\sqrt{w^2-w \sqrt{w^2-4}}}{\sqrt{2}},$$ which goes to $1$ when $w\rightarrow\infty$. The critical point $w_c$ is the smallest $w$ for which this solution exists, which is $w_c=2$.

For general $p>1$, the calculation of $w_c$ proceeds as follows. Define $u=1/w^2$ and $T=\rho^2$, then $$u=T^{1-p}-T^{-p}.$$ The critical $u_c=w_c^{-2}$ is reached when $dT/du\rightarrow\infty$ (the location of the square root singularity), hence $du/dT=0$ which gives $T=p/(p-1)\Rightarrow u_c=(p-1)^{p-1}/p^p$, and thus $$w_c=p^{p/2} (p-1)^{(1-p)/2}.$$ Check that we recover $w_c=2$ for $p=2$.