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 2HL(w)−H2L(w) between entropies on cylinders of
finite lengths L and 2L with open end cap boundaries, in the limit
L→∞. 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.Comment: 12 pages, including 17 figure