A singularity at the criticality for the free energy in percolation

Yu Zhang

Published 2003-01-15, updated 2015-10-16Version 2

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 is a sequence $\epsilon_n \downarrow 0$ such that $$\kappa''' (p_c+\epsilon_n )\leq -\epsilon_n^{-1/3+o(1)} \mbox{ and } \kappa''' (p_c-\epsilon_n )\geq \epsilon_n^{-1/3+o(1)}. $$ Note that these inequalities imply that $\kappa(p)$ is not third differentiable. This answers affirmatively a conjecture, asked by Sykes and Essam a half century ago, whether $\kappa(p)$ has a singularity at the criticality.