of equation 3.26
(here denoted by 
)
of the information correlation function 
equations 3.29
(here denoted by 
)
is given by
.
In the large n limit, where there are a large number of spins in
the system, the terms in the fraction largely cancel, leaving the simpler
result that
is 
times the mutual information between the spins (see section 6.16).
In general this mutual information and
are then
and 
differ. (Note that the logarithms are base e, and that 
appears as b in the axis label.) Thus, referring to section 6.14
for the values of the eigenvectors and eigenvalues the nth information
correlation function is given simply in this case by
and 
.
Thus, we have a straightforward interpretation of 
in the large number of spins limit (regardless of whether the external
field is zero) - it is defined on a set of k possibly widely-separated
spins, but gives us
times the mutual information between the first spin and the last spin.
The asymptotics of
in zero field are trivial, since 
.
Finally, defining 
and 
to be the probability that spins 
and 
are the same and different, respectively, we have 
,
and
Figure 6.5: Second order information correlation function of
the Ising system. Information correlation of two neighboring spins. Note
that the information correlation functions are similar up to a sign at
each order for this system. The information correlation functions are the
mutual information of the first and last spins along the chain in the k
spins considered at order k times
Figure 6.6: Third order information correlation function of
the Ising system. Information correlation of three neighboring spins.
Figure 6.7: Fourth order information correlation function of
the Ising system. Information correlation of four neighboring spins.