A derivative formula for the free energy function

TL;DR

This paper derives a formula for the derivative of the free energy function in bond percolation on the lattice, linking variance of open clusters to the derivative of the free energy.

Contribution

It introduces a derivative formula for the free energy function in bond percolation, connecting variance and the derivative of the free energy.

Findings

Variance of open clusters converges to a function involving the derivative of free energy.

Established a relationship between cluster count variance and free energy derivative.

Provided a new analytical tool for studying phase transitions in percolation.

Abstract

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 $-(p^2(1-p)+p(1-p)^2)\kappa'(p)$.