A derivative formula for the free energy function

Details A derivative formula for the free energy function A derivative formula for the free energy function

2011-08-03

arxiv.org/abs/1108.0726v1

Mathematics

Probability

8 pages 1 figure

Scientific paper

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

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