  In our previous works on infinite horizontal Ising strips of width $m$
alternating with layers of strings of Ising chains of length $n$, we found the
surprising result that the specific heats are not much different for different
values of $N$, the separation of the strings. For this reason, we study here
for $N=1$ the spin-spin correlation in the central row of each strip, and also
the central row of a strings layer. We show that these can be written as a
Toeplitz determinants. Their generating functions are ratios of two
polynomials, which in the limit of infinite vertical size become square roots
of polynomials whose degrees are $m+1$ where $m$ is the size of the strips. We
find the asymptotic behaviors near the critical temperature to be
two-dimensional Ising-like. But in regions not very close to criticality the
behavior may be different for different $m$ and $n$. Finally, in the appendix
we shall present results for generating functions in more general models.
