Percolation of randomly distributed growing clusters: Finite Size Scaling and Critical Exponents

We study the percolation properties of the growing clusters model. In this model, a number of seeds placed on random locations on a lattice are allowed to grow with a constant velocity to form clusters. When two or more clusters eventually touch each other they immediately stop their growth. The model exhibits a discontinuous transition for very low values of the seed concentration $p$ and a second, non-trivial continuous phase transition for intermediate $p$ values. Here we study in detail this continuous transition that separates a phase of finite clusters from a phase characterized by the presence of a giant component. Using finite size scaling and large scale Monte Carlo simulations we determine the value of the percolation threshold where the giant component first appears, and the critical exponents that characterize the transition. We find that the transition belongs to a different universality class from the standard percolation transition.Comment: 5 two-column pages, 6 figure

Contributions to the mixed-alkali effect in molecular dynamics simulations of alkali silicate glasses

The mixed-alkali effect on the cation dynamics in silicate glasses is analyzed via molecular dynamics simulations. Observations suggest a description of the dynamics in terms of stable sites mostly specific to one ionic species. As main contributions to the mixed--alkali slowdown longer residence times and an increased probability of correlated backjumps are identified. The slowdown is related to the limited accessibility of foreign sites. The mismatch experienced in a foreign site is stronger and more retarding for the larger ions, the smaller ions can be temporarily accommodated. Also correlations between unlike as well as like cations are demonstrated that support cooperative behavior.Comment: 10 pages, 12 figures, 1 table, revtex4, submitted to Phys. Rev.

Invaded cluster algorithm for a tricritical point in a diluted Potts model

The invaded cluster approach is extended to 2D Potts model with annealed vacancies by using the random-cluster representation. Geometrical arguments are used to propose the algorithm which converges to the tricritical point in the two-dimensional parameter space spanned by temperature and the chemical potential of vacancies. The tricritical point is identified as a simultaneous onset of the percolation of a Fortuin-Kasteleyn cluster and of a percolation of "geometrical disorder cluster". The location of the tricritical point and the concentration of vacancies for q = 1, 2, 3 are found to be in good agreement with the best known results. Scaling properties of the percolating scaling cluster and related critical exponents are also presented.Comment: 8 pages, 5 figure

Complex networks embedded in space: Dimension and scaling relations between mass, topological distance and Euclidean distance

Many real networks are embedded in space, where in some of them the links length decay as a power law distribution with distance. Indications that such systems can be characterized by the concept of dimension were found recently. Here, we present further support for this claim, based on extensive numerical simulations for model networks embedded on lattices of dimensions $d_e=1$ and $d_e=2$. We evaluate the dimension $d$ from the power law scaling of (a) the mass of the network with the Euclidean radius $r$ and (b) the probability of return to the origin with the distance $r$ travelled by the random walker. Both approaches yield the same dimension. For networks with $\delta < d_e$, $d$ is infinity, while for $\delta > 2d_e$, $d$ obtains the value of the embedding dimension $d_e$. In the intermediate regime of interest $d_e \leq \delta < 2 d_e$, our numerical results suggest that $d$ decreases continously from $d = \infty$ to $d_e$, with $d - d_e \sim (\delta - d_e)^{-1}$ for $\delta$ close to $d_e$. Finally, we discuss the scaling of the mass $M$ and the Euclidean distance $r$ with the topological distance $\ell$. Our results suggest that in the intermediate regime $d_e \leq \delta < 2 d_e$, $M(\ell)$ and $r(\ell)$ do not increase with $\ell$ as a power law but with a stretched exponential, $M(\ell) \sim \exp [A \ell^{\delta' (2 - \delta')}]$ and $r(\ell) \sim \exp [B \ell^{\delta' (2 - \delta')}]$, where $\delta' = \delta/d_e$. The parameters $A$ and $B$ are related to $d$ by $d = A/B$, such that $M(\ell) \sim r(\ell)^d$. For $\delta < d_e$, $M$ increases exponentially with $\ell$, as known for $\delta=0$, while $r$ is constant and independent of $\ell$. For $\delta \geq 2d_e$, we find power law scaling, $M(\ell) \sim \ell^{d_\ell}$ and $r(\ell) \sim \ell^{1/d_{min}}$, with $d_\ell \cdot d_{min} = d$.Comment: 17 pages, 11 figure

Detrended fluctuation analysis as a statistical tool to monitor the climate

Detrended fluctuation analysis is used to investigate power law relationship between the monthly averages of the maximum daily temperatures for different locations in the western US. On the map created by the power law exponents, we can distinguish different geographical regions with different power law exponents. When the power law exponents obtained from the detrended fluctuation analysis are plotted versus the standard deviation of the temperature fluctuations, we observe different data points belonging to the different climates, hence indicating that by observing the long-time trends in the fluctuations of temperature we can distinguish between different climates.Comment: 8 pages, 4 figures, submitted to JSTA

Global climate models violate scaling of the observed atmospheric variability

We test the scaling performance of seven leading global climate models by using detrended fluctuation analysis. We analyse temperature records of six representative sites around the globe simulated by the models, for two different scenarios: (i) with greenhouse gas forcing only and (ii) with greenhouse gas plus aerosol forcing. We find that the simulated records for both scenarios fail to reproduce the universal scaling behavior of the observed records, and display wide performance differences. The deviations from the scaling behavior are more pronounced in the first scenario, where also the trends are clearly overestimated.Comment: Accepted for publishing in Physical Review Letter

Probability Distribution of the Shortest Path on the Percolation Cluster, its Backbone and Skeleton

We consider the mean distribution functions Phi(r|l), Phi(B)(r|l), and Phi(S)(r|l), giving the probability that two sites on the incipient percolation cluster, on its backbone and on its skeleton, respectively, connected by a shortest path of length l are separated by an Euclidean distance r. Following a scaling argument due to de Gennes for self-avoiding walks, we derive analytical expressions for the exponents g1=df+dmin-d and g1B=g1S-3dmin-d, which determine the scaling behavior of the distribution functions in the limit x=r/l^(nu) much less than 1, i.e., Phi(r|l) proportional to l^(-(nu)d)x^(g1), Phi(B)(r|l) proportional to l^(-(nu)d)x^(g1B), and Phi(S)(r|l) proportional to l^(-(nu)d)x^(g1S), with nu=1/dmin, where df and dmin are the fractal dimensions of the percolation cluster and the shortest path, respectively. The theoretical predictions for g1, g1B, and g1S are in very good agreement with our numerical results.Comment: 10 pages, 3 figure

Width of percolation transition in complex networks

It is known that the critical probability for the percolation transition is not a sharp threshold, actually it is a region of non-zero width $\Delta p_c$ for systems of finite size. Here we present evidence that for complex networks $\Delta p_c \sim \frac{p_c}{l}$, where $l \sim N^{\nu_{opt}}$ is the average length of the percolation cluster, and $N$ is the number of nodes in the network. For Erd\H{o}s-R\'enyi (ER) graphs $\nu_{opt} = 1/3$, while for scale-free (SF) networks with a degree distribution $P(k) \sim k^{-\lambda}$ and $3<\lambda<4$, $\nu_{opt} = (\lambda-3)/(\lambda-1)$. We show analytically and numerically that the \textit{survivability} $S(p,l)$, which is the probability of a cluster to survive $l$ chemical shells at probability $p$, behaves near criticality as $S(p,l) = S(p_c,l) \cdot exp[(p-p_c)l/p_c]$. Thus for probabilities inside the region $|p-p_c| < p_c/l$ the behavior of the system is indistinguishable from that of the critical point

Superconductor-to-Normal Phase Transition in a Vortex Glass Model: Numerical Evidence for a New Percolation Universality Class

The three-dimensional strongly screened vortex-glass model is studied numerically using methods from combinatorial optimization. We focus on the effect of disorder strength on the ground state and found the existence of a disorder-driven normal-to-superconducting phase transition. The transition turns out to be a geometrical phase transition with percolating vortex loops in the ground state configuration. We determine the critical exponents and provide evidence for a new universality class of correlated percolation.Comment: 11 pages LaTeX using IOPART.cls, 11 eps-figures include