Mathematics – Probability
Scientific paper
2003-01-15
Mathematics
Probability
49 pages 5 figures
Scientific paper
Consider bond percolation on the square lattice and site percolation on the triangular lattice. Let $\kappa(p)$ be the free energy at the zero field. If we assume the existence of the critical exponents for the three arm and four arm paths and these critical exponents are -2/3 and -5/4, respectively, then we can show the following power law for the free energy function $\kappa(p)$: {eqnarray*} &&\kappa'''(p)= +(1/2-p)^{-1/3+\delta(|1/2-p|)}{for} p < 1/2 &&\kappa'''(p)= -(1/2-p)^{-1/3+\delta(|1/2-p|)}{for} p > 1/2, {eqnarray*} where $\delta(x)$ goes to zero as $x\to 0$. Note that the critical exponents for four arm and three arm paths indeed are proven to exist and equal -5/4 and -2/3 on the triangular lattice and the above power law for $\kappa(p)$ therefore holds for the triangular lattice. Note that the above power law for $\kappa(p)$ implies $\kappa(p)$ is not third differentiable at the critical point of the triangular lattice. This answers a long time conjecture that $\kappa(p)$ has a singularity at 1/2 since 1964 affirmatively.
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