Because of the fact that the only term in the sum for the moments that contributes is the term where all bits are one, the moments, correlation functions, and cumulants are trivial to compute. The moments of the bit string observed state distribution are given by
As before, the correlations and cumulants are identical to order three. Plots of the moments (figure 4.10), correlations (figures 4.11- 4.13) and cumulants to order four (figure 4.14) (the first three cumulants are the same as the
correlations of the same order) as a function of bit density for N=10 are
shown.
Figure 4.6: First order information correlation at N=10 as a function of q. Real part is solid. Imaginary part is dashed. This is also known as negative of the first order entropy.
Figure 4.7: Second order information correlation at N=10 as a function of q. Real part is solid. Imaginary part is dashed. This is also known as the mutual information.
Figure 4.8: Third order information correlation at N=10 as a function of q. Real part is solid. Imaginary part is dashed.
Figure 4.9: Fourth order information correlation at N=10 as a function of q. Real part is solid. Imaginary part is dashed.
Figure 4.10: Moments 1 thru 4 at N=10 as a function of q. Increasing order moments are more curved.
Figure 4.11: Second order correlation at N=10 as a function of q. This is also the second order cumulant.
Figure 4.12: Third order correlation at N=10 as a function of q. This is also the third order cumulant.
Figure 4.13: Fourth order correlation at N=10 as a function of q.
Figure 4.14: Fourth order cumulant at N=10 as a function of q.