A derivative formula for the free energy function

Abstract

We consider bond percolation on the ${\mathbf{Z}}^d$ lattice. Let $M_n$ be the number of open clusters in $B(n)=[-n,n{]}^d$. It is well known that $E_p{M}_n/(2n+1{)}^d$ converges to the free energy function $\kappa (p)$ at the zero field. In this paper, we show that ${\sigma }_p^2(M_n)/(2n+1{)}^d$ converges to $-(p^2(1-p)+p(1-p{)}^2){\kappa }^{\prime }(p)$.

8 pages 1 figure

Cited by in corpus (1)

• A lower bound on the two-arms exponent for critical percolation on the lattice