A singularity at the criticality for the free energy in percolation

Abstract

Consider percolation on the triangular lattice. Let $\kappa(p)$ be the free energy at the zero field. We show that $$|\kappa'''(p)| \leq |p-p_c|^{-1/3+o(1)} \mbox{ if } p \neq p_c.$$ Furthermore, we show that there exists a sequence $\epsilon_n\downarrow 0$ such that $$|\kappa'''(p_c\pm \epsilon_n)|\geq \epsilon_n^{-1/3+o(1)}.$$ This answers affirmatively a conjecture, asked by Sykes and Essam a half century ago, whether $\kappa(p)$ has a singularity at the criticality.