From equation 6.35, the moments of k spins in the Ising system are given by
In the thermodynamic limit this simplifies as
where
From the moments, the correlations and cumulants may be calculated. Figures 6.8- 6.11 show moments of orders 1 through 4 of neighboring spins. Figures 6.12- 6.14 show the correlation functions of orders 2 through 4 of neighboring spins. Figure 6.15 shows the cumulant function of orders 4 of neighboring spins. Recall that the first three correlation and cumulant functions are equal by order.
Figure 6.8: First order moment - moment of one spin. Note the transitional behavior in the region of the increase in the entropy, where the states 
becomes dominated by 
.
Figure 6.9: Second order moment of two neighboring spins.
Figure 6.10: Third order moment of three neighboring spins.
Figure 6.11: Fourth order moment of four neighboring spins.
Figure 6.12: Second order correlation of two neighboring spins.
Figure 6.13: Third order correlation of three neighboring spins.
Figure 6.14: Fourth order correlation of four neighboring spins.
Figure 6.15: Fourth order cumulant of four neighboring spins.