The vertex-cover problem on the Hanoi networks HN3 and HN5 is analyzed with an exact renormalization group and parallel-tempering Monte Carlo simulations. The grand canonical partition function of the equivalent hard-core repulsive lattice-gas proble
m is recast first as an Ising-like canonical partition function, which allows for a closed set of renormalization group equations. The flow of these equations is analyzed for the limit of infinite chemical potential, at which the vertex-cover problem is attained. The relevant fixed point and its neighborhood are analyzed, and non-trivial results are obtained both, for the coverage as well as for the ground state entropy density, which indicates the complex structure of the solution space. Using special hierarchy-dependent operators in the renormalization group and Monte-Carlo simulations, structural details of optimal configurations are revealed. These studies indicate that the optimal coverages (or packings) are not related by a simple symmetry. Using a clustering analysis of the solutions obtained in the Monte Carlo simulations, a complex solution space structure is revealed for each system size. Nevertheless, in the thermodynamic limit, the solution landscape is dominated by one huge set of very similar solutions.
The far-from-equilibrium dynamics of glassy systems share important phenomenological traits. A transition is generally observed from a time-homogeneous dynamical regime to an aging regime where physical changes occur intermittently and, on average, a
t a decreasing rate. It has been suggested that a global change of the independent time variable to its logarithm may render the aging dynamics homogeneous: for colloids, this entails diffusion but on a logarithmic time scale. Our novel analysis of experimental colloid data confirms that the mean square displacement grows linearly in time at low densities and shows that it grows linearly in the logarithm of time at high densities. Correspondingly, pairs of particles initially in close contact survive as pairs with a probability which decays exponentially in either time or its logarithm. The form of the Probability Density Function of the displacements shows that long-ranged spatial correlations are very long-lived in dense colloids. A phenomenological stochastic model is then introduced which relies on the growth and collapse of strongly correlated clusters (dynamic heterogeneity), and which reproduces the full spectrum of observed colloidal behaviors depending on the form assumed for the probability that a cluster collapses during a Monte Carlo update. In the limit where large clusters dominate, the collapse rate is ~1/t, implying a homogeneous, log-Poissonian process that qualitatively reproduces the experimental results for dense colloids. Finally an analytical toy-model is discussed to elucidate the strong dependence of the simulation results on the integrability (or lack thereof) of the cluster collapse probability function.
Two new classes of networks are introduced that resemble small-world properties. These networks are recursively constructed but retain a fixed, regular degree. They consist of a one-dimensional lattice backbone overlayed by a hierarchical sequence of
long-distance links. Both types of networks, one 3-regular and the other 4-regular, lead to distinct behaviors, as revealed by renormalization group studies. The 3-regular networks are planar, have a diameter growing as sqrt{N} with the system size N, and lead to super-diffusion with an exact, anomalous exponent d_w=1.3057581..., but possesses only a trivial fixed point T_c=0 for the Ising ferromagnet. In turn, the 4-regular networks are non-planar, have a diameter growing as ~2^[sqrt(log_2 N^2)], exhibit ballistic diffusion (d_w=1), and a non-trivial ferromagnetic transition, T_c>0. It suggest that the 3-regular networks are still quite geometric, while the 4-regular networks qualify as true small-world networks with mean-field properties. As an example of an application we discuss synchronization of processors on these networks.
Numerical results for the local field distributions of a family of Ising spin-glass models are presented. In particular, the Edwards-Anderson model in dimensions two, three, and four is considered, as well as spin glasses with long-range power-law-mo
dulated interactions that interpolate between a nearest-neighbour Edwards-Anderson system in one dimension and the infinite-range Sherrington-Kirkpatrick model. Remarkably, the local field distributions only depend weakly on the range of the interactions and the dimensionality, and show strong similarities except for near zero local field.
