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 $$|κ'''(p)| \leq |p-p_c|^{-1/3+o(1)} \mbox{ if } p \neq p_c.$$ Furthermore, we show that there exists a sequence ${\epsilon }_n\downarrow 0$ such that $$|{\kappa }^{\prime \prime \prime }(p_c\pm {\epsilon }_n)|\ge {\epsilon }_n^{-1/3+o(1)}.$$ This answers affirmatively a conjecture, asked by Sykes and Essam a half century ago, whether $\kappa (p)$ has a singularity at the criticality.

38 pages 4 figures

References in corpus (1)

• Path Crossing Exponents and the External Perimeter in 2D Percolation