Discontinuous percolation transition with full step order parameter jump

We demonstrate that there is a class of discontinuous percolation that is characterized by full step jump in order parameter at threshold $p_c=1$. Such a percolation takes place in the infinite dimension if the critical exponent $\tau$, the decay exponent of the cluster number density distribution in critical regime, holds $1<\tau \le 2$. The scaling relations of $\sigma = 2-\tau$ and $\gamma = 1/\sigma$ are derived for the critical exponents $\sigma$ and $\gamma$ associated with the characteristic cluster size and with the susceptibility, respectively. We also show that the cluster number density distribution is compact and can be widened up to $\sim \sqrt{N}$ for system size $N$.