Extended droplet theory for aging in short-ranged spin glasses and a numerical examination

We analyze isothermal aging of a four dimensional Edwards-Anderson model in detail by Monte Carlo simulations. We analyze the data in the view of an extended version of the droplet theory proposed recently (cond-mat/0202110) which is based on the original droplet theory plus conjectures on the anomalously soft droplets in the presence of domain walls. We found that the scaling laws including some fundamental predictions of the original droplet theory explain well our results. The results of our simulation strongly suggest the separation of the breaking of the time translational invariance and the fluctuation dissipation theorem in agreement with our scenario.Comment: 27 pages, 39 epsfiles, revised versio

Scaling Properties of Random Walks on Small-World Networks

Using both numerical simulations and scaling arguments, we study the behavior of a random walker on a one-dimensional small-world network. For the properties we study, we find that the random walk obeys a characteristic scaling form. These properties include the average number of distinct sites visited by the random walker, the mean-square displacement of the walker, and the distribution of first-return times. The scaling form has three characteristic time regimes. At short times, the walker does not see the small-world shortcuts and effectively probes an ordinary Euclidean network in $d$-dimensions. At intermediate times, the properties of the walker shows scaling behavior characteristic of an infinite small-world network. Finally, at long times, the finite size of the network becomes important, and many of the properties of the walker saturate. We propose general analytical forms for the scaling properties in all three regimes, and show that these analytical forms are consistent with our numerical simulations.Comment: 7 pages, 8 figures, two-column format. Submitted to PR

XY model in small-world networks

The phase transition in the XY model on one-dimensional small-world networks is investigated by means of Monte-Carlo simulations. It is found that long-range order is present at finite temperatures, even for very small values of the rewiring probability, suggesting a finite-temperature transition for any nonzero rewiring probability. Nature of the phase transition is discussed in comparison with the globally-coupled XY model.Comment: 5 pages, accepted in PR

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.

Simple models of small world networks with directed links

We investigate the effect of directed short and long range connections in a simple model of small world network. Our model is such that we can determine many quantities of interest by an exact analytical method. We calculate the function $V(T)$, defined as the number of sites affected up to time $T$ when a naive spreading process starts in the network. As opposed to shortcuts, the presence of un-favorable bonds has a negative effect on this quantity. Hence the spreading process may not be able to affect all the network. We define and calculate a quantity named the average size of accessible world in our model. The interplay of shortcuts, and un-favorable bonds on the small world properties is studied.Comment: 15 pages, 9 figures, published versio

Relaxation Properties of Small-World Networks

Recently, Watts and Strogatz introduced the so-called small-world networks in order to describe systems which combine simultaneously properties of regular and of random lattices. In this work we study diffusion processes defined on such structures by considering explicitly the probability for a random walker to be present at the origin. The results are intermediate between the corresponding ones for fractals and for Cayley trees.Comment: 16 pages, 6 figure

Glassiness and constrained dynamics of a short-range non-disordered spin model

We study the low temperature dynamics of a two dimensional short-range spin system with uniform ferromagnetic interactions, which displays glassiness at low temperatures despite the absence of disorder or frustration. The model has a dual description in terms of free defects subject to dynamical constraints, and is an explicit realization of the ``hierarchically constrained dynamics'' scenario for glassy systems. We give a number of exact results for the statics of the model, and study in detail the dynamical behaviour of one-time and two-time quantities. We also consider the role played by the configurational entropy, which can be computed exactly, in the relation between fluctuations and response.Comment: 10 pages, 9 figures; minor changes, references adde

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

Synchronization, Diversity, and Topology of Networks of Integrate and Fire Oscillators

We study synchronization dynamics of a population of pulse-coupled oscillators. In particular, we focus our attention in the interplay between networks topological disorder and its synchronization features. Firstly, we analyze synchronization time $T$ in random networks, and find a scaling law which relates $T$ to networks connectivity. Then, we carry on comparing synchronization time for several other topological configurations, characterized by a different degree of randomness. The analysis shows that regular lattices perform better than any other disordered network. The fact can be understood by considering the variability in the number of links between two adjacent neighbors. This phenomenon is equivalent to have a non-random topology with a distribution of interactions and it can be removed by an adequate local normalization of the couplings.Comment: 6 pages, 8 figures, LaTeX 209, uses RevTe

Ultra-Slow Vacancy-Mediated Tracer Diffusion in Two Dimensions: The Einstein Relation Verified

We study the dynamics of a charged tracer particle (TP) on a two-dimensional lattice all sites of which except one (a vacancy) are filled with identical neutral, hard-core particles. The particles move randomly by exchanging their positions with the vacancy, subject to the hard-core exclusion. In case when the charged TP experiences a bias due to external electric field ${\bf E}$, (which favors its jumps in the preferential direction), we determine exactly the limiting probability distribution of the TP position in terms of appropriate scaling variables and the leading large-N ($n$ being the discrete time) behavior of the TP mean displacement $\bar{{\bf X}}_n$; the latter is shown to obey an anomalous, logarithmic law $|\bar{{\bf X}}_n| = \alpha_0(|{\bf E}|) \ln(n)$. On comparing our results with earlier predictions by Brummelhuis and Hilhorst (J. Stat. Phys. {\bf 53}, 249 (1988)) for the TP diffusivity $D_n$ in the unbiased case, we infer that the Einstein relation $\mu_n = \beta D_n$ between the TP diffusivity and the mobility $\mu_n = \lim_{|{\bf E}| \to 0}(|\bar{{\bf X}}_n|/| {\bf E} |n)$ holds in the leading in $n$ order, despite the fact that both $D_n$ and $\mu_n$ are not constant but vanish as $n \to \infty$. We also generalize our approach to the situation with very small but finite vacancy concentration $\rho$, in which case we find a ballistic-type law $|\bar{{\bf X}}_n| = \pi \alpha_0(|{\bf E}|) \rho n$. We demonstrate that here, again, both $D_n$ and $\mu_n$, calculated in the linear in $\rho$ approximation, do obey the Einstein relation.Comment: 25 pages, one figure, TeX, submitted to J. Stat. Phy