Finite-size scaling of correlation functions in finite systems

We propose the finite-size scaling of correlation functions in finite systems near their critical points. At a distance r in a d-dimensional finite system of size L, the correlation function can be written as the product of |r|^{−(d−2+η)} and a finite-size scaling function of the variables r/L and tL^1/v, where t = (T − T_c)=T_c, η is the critical exponent of correlation function, and v is the critical exponent of correlation length. The correlation function only has a sigificant directional dependence when |r| is compariable to L. We then confirm this finite-size scaling by calculating the correlation functions of the two-dimensional Ising model and the bond percolation in two-dimensional lattices using Monte Carlo simulations. We can use the finite-size scaling of the correlation function to determine the critical point and the critical exponent η.