# Proving the relation between order parameter and its exponent

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, $$$$O \sim a\lvert T - T_c \rvert^\beta$$$$ where $$a$$ is some numerical value. In my case, the order parameter is $$\Delta\lambda$$, hence, $$$$\Delta\lambda \sim a\lvert T - T_c \rvert^\beta$$$$

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, $$$$\lambda(r_h) = \lambda_c + \left[ \frac{\partial\lambda}{\partial r_h}\right]_c \left(r_h - r_c\right) + \mathcal{O}(r_h)$$$$

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, $$$$r_p = r_c\left(1 + \Delta\right)\,,$$$$ with $$\lvert\Delta\rvert \ll 1$$.

Similarly, the Hawking temperature $$T$$ at the phase transition point can be written as, $$$$T_p = T_c \left(1+\epsilon\right)\,,$$$$ where $$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, $$$$\label{eq:taylor-expand-temp} T(r_h) = T_c + \left[ \frac{\partial T}{\partial r_h}\right]_c \left(r_h - r_c\right) + \frac{1}{2}\left[ \frac{\partial^2 T}{\partial {r_h}^2}\right]_c \left(r_h - r_c\right)^2 + \mathcal{O}(r_h) \,,$$$$

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!