Article

# Cluster aggregation model for discontinuous percolation transitions

Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea.
(Impact Factor: 2.29). 03/2010; 81(3 Pt 1):030103. DOI: 10.1103/PhysRevE.81.030103
Source: PubMed

ABSTRACT

The evolution of the Erdos-Rényi (ER) network by adding edges is a basis model for irreversible kinetic aggregation phenomena. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel Kij approximately ij , where ij is the product of the sizes of two merging clusters. Here we study that when the giant cluster is discouraged to develop by a sublinear kernel Kij approximately (ij)omega with 0<or=omega<1/2 , the percolation transition (PT) is discontinuous. Such discontinuous PT can occur even when the ER dynamics evolves from proper initial conditions. The obtained evolutionary properties of the simple model sheds light on the origin of the discontinuous PT in other nonequilibrium kinetic systems.

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• "This observation was confirmed in a number of subsequent works based on simulations, including Refs. [8] [9] [10] [11] [12] [13]. In our work [14], we showed that the conclusions for the local optimization based models obtained from these simulations were incorrect, and the so-called explosive percolation transition is actually continuous for infinite systems. "
##### Article: Critical exponents of the explosive percolation transition
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ABSTRACT: In a new type of percolation phase transition, which was observed in a set of non-equilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptas-like algorithm. This causes preferential merging of small components and delays the emergence of the percolation cluster. First simulations led to a conclusion that a percolation cluster in this irreversible process is born discontinuously, by a discontinuous phase transition, which results in the term "explosive percolation transition". We have shown that this transition is actually continuous (second-order) though with anomalously small critical exponent of the percolation cluster. Here we propose an efficient numerical method enabling us to find the critical exponents and other characteristics of this second order transition for a representative set of explosive percolation models with different number of choices. The method is based on sewing together the numerical solutions of evolution equations for the cluster size distribution and power-law asymptotics. For each of the models, with high precision, we obtain critical exponents and the critical point.
Full-text · Article · Apr 2014 · Physical Review E
• ##### Article: Self-guiding and red shifts of intense pulses propagating in clustered gases
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ABSTRACT: Summary form only given. Intense laser interaction with clusters, Van der Waals-bonded aggregates of up to ∼107 atoms, is of great interest owing to applications which include the generation of X-rays, energetic ions and electrons, and nuclear particles. In recent work, the time-resolved explosion of intense-laser-heated clusters has been observed, and this explosion scenario leads to intense laser self-focusing in a clustered gas. Here, we report measurements of the frequency shifts of intense laser pulses propagating in clustered gases. This measurement is consistent with our previous observations and model. We use a pump-probe setup for the frequency shift measurement. For a series of time-delayed 2ω probe (sub-80 fs, 399 nm) spectra modulated by an intense pump (1.5 mJ, 80 fs, 798 nm) pulse co-propagating in a gas of argon clusters, we find that the probe increasingly red-shifts during the phase where the pump self-focuses. At later times, the shift goes blue and then relaxes back to zero. The red-shifts at early delays indicate the positive transients of the refractive index of pump-heated clusters, in agreement with our recent observation of self-focusing. This red-shift strongly contrasts with the blue-shift commonly observed in the ionization of non-clustered gases. At later times, however, the probe frequency shift becomes blue and then relaxes to zero owing to the dominant response of the underdense portion of the cluster plasmas at later times, with that response becoming progressively weaker.
No preview · Article · Jan 2004 · IEEE International Conference on Plasma Science
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##### Article: A new route to Explosive Percolation
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ABSTRACT: The biased link occupation rule in the Achlioptas process (AP) discourages the large clusters to grow much ahead of others and encourages faster growth of clusters which lag behind. In this paper we propose a model where this tendency is sharply reflected in the Gamma distribution of the cluster sizes, unlike the power law distribution in AP. In this model single edges between pairs of clusters of sizes $s_i$ and $s_j$ are occupied with a probability $\propto (s_is_j)^{\alpha}$. The parameter $\alpha$ is continuously tunable over the entire real axis. Numerical studies indicate that for $\alpha < \alpha_c$ the transition is first order, $\alpha_c=0$ for square lattice and $\alpha_c=-1/2$ for random graphs. In the limits of $\alpha = -\infty, +\infty$ this model coincides with models well established in the literature.
Full-text · Article · Nov 2009 · Physica A: Statistical Mechanics and its Applications