Self-avoiding walks and connective constants in small-world networks

Long-distance characteristics of small-world networks have been studied by means of self-avoiding walks (SAW's). We consider networks generated by rewiring links in one- and two-dimensional regular lattices. The number of SAW's $u_n$ was obtained from numerical simulations as a function of the number of steps $n$ on the considered networks. The so-called connective constant, $\mu = \lim_{n \to \infty} u_n/u_{n-1}$, which characterizes the long-distance behavior of the walks, increases continuously with disorder strength (or rewiring probability, $p$). For small $p$, one has a linear relation $\mu = \mu_0 + a p$, $\mu_0$ and $a$ being constants dependent on the underlying lattice. Close to $p = 1$ one finds the behavior expected for random graphs. An analytical approach is given to account for the results derived from numerical simulations. Both methods yield results agreeing with each other for small $p$, and differ for $p$ close to 1, because of the different connectivity distributions resulting in both cases.Comment: 7 pages, 5 figure

Interface Motion and Pinning in Small World Networks

We show that the nonequilibrium dynamics of systems with many interacting elements located on a small-world network can be much slower than on regular networks. As an example, we study the phase ordering dynamics of the Ising model on a Watts-Strogatz network, after a quench in the ferromagnetic phase at zero temperature. In one and two dimensions, small-world features produce dynamically frozen configurations, disordered at large length scales, analogous of random field models. This picture differs from the common knowledge (supported by equilibrium results) that ferromagnetic short-cuts connections favor order and uniformity. We briefly discuss some implications of these results regarding the dynamics of social changes.Comment: 4 pages, 5 figures with minor corrections. To appear in Phys. Rev.

Superconductor-insulator quantum phase transition in a single Josephson junction

The superconductor-to-insulator quantum phase transition in resistively shunted Josephson junctions is investigated by means of path-integral Monte Carlo simulations. This numerical technique allows us to directly access the (previously unexplored) regime of the Josephson-to-charging energy ratios E_J/E_C of order one. Our results unambiguously support an earlier theoretical conjecture, based on renormalization-group calculations, that at T -> 0 the dissipative phase transition occurs at a universal value of the shunt resistance R_S = h/4e^2 for all values E_J/E_C. On the other hand, finite-temperature effects are shown to turn this phase transition into a crossover, which position depends significantly on E_J/E_C, as well as on the dissipation strength and on temperature. The latter effect needs to be taken into account in order to reconcile earlier theoretical predictions with recent experimental results.Comment: 7 pages, 6 figure

Aharonov-Bohm oscillations of a particle coupled to dissipative environments

The amplitude of the Bohm-Aharonov oscillations of a particle moving around a ring threaded by a magnetic flux and coupled to different dissipative environments is studied. The decay of the oscillations when increasing the radius of the ring is shown to depend on the spatial features of the coupling. When the environment is modelled by the Caldeira-Leggett bath of oscillators, or the particle is coupled by the Coulomb potential to a dirty electron gas, interference effects are suppressed beyond a finite length, even at zero temperature. A finite renormalization of the Aharonov-Bohm oscillations is found for other models of the environment.Comment: 6 page

Two-loop approximation in the Coulomb blockade problem

We study Coulomb blockade (CB) oscillations in the thermodynamics of a metallic grain which is connected to a lead by a tunneling contact with a large conductance $g_0$ in a wide temperature range, $E_Cg_0^4 e^{-g_0/2}<T<E_C$, where $E_C$ is the charging energy. Using the instanton analysis and the renormalization group we obtain the temperature dependence of the amplitude of CB oscillations which differs from the previously obtained results. Assuming that at $T < E_Cg_0^4 e^{-g_0/2}$ the oscillation amplitude weakly depends on temperature we estimate the magnitude of CB oscillations in the ground state energy as $E_Cg_0^4 e^{-g_0/2}$.Comment: 10 pages, 3 figure

Nonequilibrium transitions in complex networks: a model of social interaction

We analyze the non-equilibrium order-disorder transition of Axelrod's model of social interaction in several complex networks. In a small world network, we find a transition between an ordered homogeneous state and a disordered state. The transition point is shifted by the degree of spatial disorder of the underlying network, the network disorder favoring ordered configurations. In random scale-free networks the transition is only observed for finite size systems, showing system size scaling, while in the thermodynamic limit only ordered configurations are always obtained. Thus in the thermodynamic limit the transition disappears. However, in structured scale-free networks, the phase transition between an ordered and a disordered phase is restored.Comment: 7 pages revtex4, 10 figures, related material at http://www.imedea.uib.es/PhysDept/Nonlinear/research_topics/Social

Transport Properties of Random Walks on Scale-Free/Regular-Lattice Hybrid Networks

We study numerically the mean access times for random walks on hybrid disordered structures formed by embedding scale-free networks into regular lattices, considering different transition rates for steps across lattice bonds ($F$) and across network shortcuts ($f$). For fast shortcuts ($f/F\gg 1$) and low shortcut densities, traversal time data collapse onto an universal curve, while a crossover behavior that can be related to the percolation threshold of the scale-free network component is identified at higher shortcut densities, in analogy to similar observations reported recently in Newman-Watts small-world networks. Furthermore, we observe that random walk traversal times are larger for networks with a higher degree of inhomogeneity in their shortcut distribution, and we discuss access time distributions as functions of the initial and final node degrees. These findings are relevant, in particular, when considering the optimization of existing information networks by the addition of a small number of fast shortcut connections.Comment: 8 pages, 6 figures; expanded discussions, added figures and references. To appear in J Stat Phy

Comparison of two non-primitive methods for path integral simulations: Higher-order corrections vs. an effective propagator approach

Two methods are compared that are used in path integral simulations. Both methods aim to achieve faster convergence to the quantum limit than the so-called primitive algorithm (PA). One method, originally proposed by Takahashi and Imada, is based on a higher-order approximation (HOA) of the quantum mechanical density operator. The other method is based upon an effective propagator (EPr). This propagator is constructed such that it produces correctly one and two-particle imaginary time correlation functions in the limit of small densities even for finite Trotter numbers P. We discuss the conceptual differences between both methods and compare the convergence rate of both approaches. While the HOA method converges faster than the EPr approach, EPr gives surprisingly good estimates of thermal quantities already for P = 1. Despite a significant improvement with respect to PA, neither HOA nor EPr overcomes the need to increase P linearly with inverse temperature. We also derive the proper estimator for radial distribution functions for HOA based path integral simulations.Comment: 17 pages, latex, 6 postscript figure

Introducing Small-World Network Effect to Critical Dynamics

We analytically investigate the kinetic Gaussian model and the one-dimensional kinetic Ising model on two typical small-world networks (SWN), the adding-type and the rewiring-type. The general approaches and some basic equations are systematically formulated. The rigorous investigation of the Glauber-type kinetic Gaussian model shows the mean-field-like global influence on the dynamic evolution of the individual spins. Accordingly a simplified method is presented and tested, and believed to be a good choice for the mean-field transition widely (in fact, without exception so far) observed on SWN. It yields the evolving equation of the Kawasaki-type Gaussian model. In the one-dimensional Ising model, the p-dependence of the critical point is analytically obtained and the inexistence of such a threshold p_c, for a finite temperature transition, is confirmed. The static critical exponents, gamma and beta are in accordance with the results of the recent Monte Carlo simulations, and also with the mean-field critical behavior of the system. We also prove that the SWN effect does not change the dynamic critical exponent, z=2, for this model. The observed influence of the long-range randomness on the critical point indicates two obviously different hidden mechanisms.Comment: 30 pages, 1 ps figures, REVTEX, accepted for publication in Phys. Rev.

The Standard Model as a noncommutative geometry: the low energy regime

We render a thorough, physicist's account of the formulation of the Standard Model (SM) of particle physics within the framework of noncommutative differential geometry (NCG). We work in Minkowski spacetime rather than in Euclidean space. We lay the stress on the physical ideas both underlying and coming out of the noncommutative derivation of the SM, while we provide the necessary mathematical tools. Postdiction of most of the main characteristics of the SM is shown within the NCG framework. This framework, plus standard renormalization technique at the one-loop level, suggest that the Higgs and top masses should verify 1.3 m_top \lesssim m_H \lesssim 1.73 m_top.Comment: 44 pages, Plain TeX with AMS fonts, mass formulae readjusted, some references added, to appear in Physics Report