number of vertices in the core $N_c$. The average connectivity of the core is $$\alpha_{eff} = \frac{2l(\alpha)}{c(\alpha)} = \frac{B-A}{1-A}=2+
 \frac{2(6)^{1/2}}{3}\varepsilon ^{1/2}+\frac{4}{3}\varepsilon
 +O(\varepsilon^{3 /2})$$ for $\varepsilon > 0$, which implies that giant core component should look like a large loop for $\alpha$ close to $e^+$. The exponent $1/2$ in the first correction makes such a property quite difficult to see numerically at large but finite $N$.

In the random graph model, the vertices do not live in any ambient space, and the notion of correlation length is ambiguous. This problem will reappear in the finite size scaling analysis of the next section. However, the emergence of the core is very reminiscent of critical phenomena in physics. In particular, the critical slowing down is observable during the global leaf removal process. Indeed, the speed of convergence of the iterated exponential sequence can be computed. One finds that for $\alpha
\neq e$, the convergence is exponential: the convergence rate $\xi^{-1}(\alpha)$ is given by the formula $$\xi^{-1} =\frac{A+B}{2}-\log \alpha,$$ and more precisely $$\begin{aligned}
 e_{n}-\frac{W}{\alpha} & \sim & (-)^n w e^{-n/\xi} \quad
 \mathrm{for} \; \alpha <e, \\ 
 e_{2n+1}-\frac{A}{\alpha} & \sim & - a e^{-(2n+1)/\xi} \quad
 \mathrm{for} \; \alpha >e, \\
 e_{2n}-\frac{B}{\alpha} & \sim & b 