e^{-2n/\xi} \quad
 \mathrm{for} \; \alpha >e,\end{aligned}$$ where $a(\alpha)$, $w(\alpha)$ and $b(\alpha)$ are positive functions (they coincide for $\alpha < e$ and $be^{-B/2}=ae^{-A/2}$). When $\alpha \to
e^-$, $\xi \sim \frac{2e}{e-\alpha}$, and when $\alpha \to e^+$, $\xi \sim
\frac{e}{\alpha-e}$.

At $\alpha=e$, the convergence is algebraic, and $$\begin{aligned}
 \lefteqn{ e \left(e_n-\frac{1}{e}\right) = } \\
 & & (-)^n\frac{6^{1/2}}{n^{1/2}}+\frac{3}{2n}+(-)^{n+1} 
     \frac{21(6)^{1/2}}{80} \frac{\log n}{n^{3/2}} 
     +O(\frac{1}{n^{3/2}}).\end{aligned}$$

This leads to the asymptotics at $\alpha=e$: $$\begin{aligned}
 i_n & = & \frac{3-e}{e}+O\left(\frac{1}{n^{3/2}}\right) , \\
 c_n & = & \frac{6}{en}\left(1-\frac{3^{1/2}}{4n^{1/2}}-\frac{21}{80}
 \frac{\log n}{n} +O\left(\frac{1}{n}\right)\right) , \\
 l_n & = & \frac{6}{en}\left(1-\frac{21}{80}\frac{\log n}{n}
 +O\left(\frac{1}{n}\right)\right).\end{aligned}$$ The first correction for $c_n$ is more important that the one for $l_n$. Moreover the logarithms at $\alpha=e$ lead to suspect that the finite size analysis of the next section might also be complicated by logarithms.

Numerical studies of the core percolation {#sec:numsim}
=========================================

The analytical computations above have enabled us to locate a phase transition at $\alpha = e$. They give information concerning the critical region but do not exhaust all the critical exponents. So we made an extensive numerical analysis of the core using Monte-Carlo simulations. At the first step this can also be used to check the previous analytical results. But let us start with the numerical 