  We present numerical results for various information theoretic properties of
the square lattice Ising model. First, using a bond propagation algorithm, we
find the difference $2H_L(w) - H_{2L}(w)$ between entropies on cylinders of
finite lengths $L$ and 2L with open end cap boundaries, in the limit
$L\to\infty$. This essentially quantifies how the finite length correction for
the entropy scales with the cylinder circumference $w$. Secondly, using the
transfer matrix, we obtain precise estimates for the information needed to
specify the spin state on a ring encircling an infinite long cylinder.
Combining both results we obtain the mutual information between the two halves
of a cylinder (the "excess entropy" for the cylinder), where we confirm with
higher precision but for smaller systems results recently obtained by Wilms et
al. -- and we show that the mutual information between the two halves of the
ring diverges at the critical point logarithmically with $w$. Finally we use
the second result together with Monte Carlo simulations to show that also the
excess entropy of a straight line of $n$ spins in an infinite lattice diverges
at criticality logarithmically with $n$. We conjecture that such logarithmic
divergence happens generically for any one-dimensional subset of sites at any
2-dimensional second order phase transition. Comparing straight lines on square
and triangular lattices with square loops and with lines of thickness 2, we
discuss questions of universality.
