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My main goal is to reproduce equation ($57$) of this paper, for which I use the following method:

The general expression which relates any order parameter $O$ by its critical exponent $\beta$ is given by, \begin{equation} O \sim a\lvert T - T_c \rvert^\beta \end{equation} where $a$ is some numerical value. In my case, the order parameter is $\Delta\lambda$, hence, \begin{equation} \Delta\lambda \sim a\lvert T - T_c \rvert^\beta \end{equation}

To achieve this kind of relation, First, I do a Taylor expansion of $\lambda(r_h)$ up to first order around the critical point as, \begin{equation} \lambda(r_h) = \lambda_c + \left[ \frac{\partial\lambda}{\partial r_h}\right]_c \left(r_h - r_c\right) + \mathcal{O}(r_h) \end{equation}

Also, the relation between the horizon radius at the phase transition point $r_p$ and the horizon radius at the critical point $r_c$ is given by, \begin{equation} r_p = r_c\left(1 + \Delta\right)\,, \end{equation} with $\lvert\Delta\rvert \ll 1$.

Similarly, the Hawking temperature $T$ at the phase transition point can be written as, \begin{equation} T_p = T_c \left(1+\epsilon\right)\,, \end{equation} where $T_c$ is the critical Temperature and $\lvert\epsilon\rvert \ll 1$.

Again, I evaluate the Taylor expansion of $T(r_h)$ up to the second order around the critical point as, \begin{equation}\label{eq:taylor-expand-temp} T(r_h) = T_c + \left[ \frac{\partial T}{\partial r_h}\right]_c \left(r_h - r_c\right) + \frac{1}{2}\left[ \frac{\partial^2 T}{\partial {r_h}^2}\right]_c \left(r_h - r_c\right)^2 + \mathcal{O}(r_h) \,, \end{equation}

Now, how can I prove that $\frac{\Delta\lambda}{\lambda_c} = k \sqrt{\frac{T}{T_c}-1}$ using above equations? Where $\Delta\lambda = \lambda_s - \lambda_l$, i.e. difference of the $\lambda$ values for small and large black holes calculated at the transition point.

Any help in this regard would be truly beneficial!

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