Information theoretic aspects of the two-dimensional Ising model

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 infinitely 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 the 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 two-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.