A singularity at the criticality for the free energy in percolation

Consider percolation on the triangular lattice. Let $\kappa (p)$ be the free energy at the zero field. We show that $$\mathrm{\mid }{\kappa }^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(p)\mathrm{\mid }\le \mathrm{\mid }p-p_c{\mathrm{\mid }}^{-1\mathrm{/}3+o(1)}ifp\mathrm{\ne }{p}_c\mathrm{.}$$ Furthermore, we show that there exists a sequence $\epsilon_n\downarrow 0$suchthat$$\mathrm{\mid }{\kappa }^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}(p_c\pm {ϵ}_n)\mathrm{\mid }\ge {ϵ}_n^{-1\mathrm{/}3+o(1)}\mathrm{.}$$ This answers affirmatively a conjecture, asked by Sykes and Essam a half century ago, whether $\kappa (p)$ has a singularity at the criticality.