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.

Ordinary bond percolation (OP) can be viewed as a process where clusters grow by joining them pairwise, by adding links chosen randomly one by one from a set of predefined 'virtual' links. In contrast, in agglomerative percolation (AP) clusters grow by choosing randomly a 'target cluster' and joining it with all its neighbors, as defined by the same set of virtual links. Previous studies showed that AP is in different universality classes from OP for several types of (virtual) networks (linear chains, trees, Erdös-Rényi networks), but most surprising were the results for 2-d lattices: While AP on the triangular lattice was found to be in the OP universality class, it behaved completely differently on the square lattice. In the present paper we explain this striking violation of universality by invoking bipartivity. While the square lattice is a bipartite graph, the triangular lattice is not. In conformity with this we show that AP on the honeycomb and simple cubic (3-d) lattices -both of which are bipartite -are also not in the OP universality classes. More precisely, we claim that this violation of universality is basically due to a Z2 symmetry that is spontaneously broken at the percolation threshold. We also discuss AP on bipartite random networks and suitable generalizations of AP on k−partite graphs.

PACS 05.40.Fb -Random walks and Levy flights PACS 89.75.Hc -Networks and genealogical trees PACS 87.10.-e -General theory and mathematical aspectsAbstract. -The first passage time (FPT) distribution for random walk in complex networks is calculated through an asymptotic analysis. For network with size N and short relaxation time τ N , the computed mean first passage time (MFPT), which is inverse of the decay rate of FPT distribution, is inversely proportional to the degree of the destination. These results are verified numerically for the paradigmatic networks with excellent agreement. We show that the range of validity of the analytical results covers networks that have short relaxation time and high mean degree, which turn out to be valid to many real networks.

We propose a method to create "spin cat states," i.e., macroscopic superpositions of coherent spin states, in Bose-Einstein condensates using the Kerr nonlinearity due to atomic collisions. Based on a detailed study of atom loss, we conclude that cat sizes of hundreds of atoms should be realistic. The existence of the spin cat states can be demonstrated by optical readout. Our analysis also includes the effects of higher-order nonlinearities, atom number fluctuations, and limited readout efficiency.