Persistence in cluster-cluster aggregation

Complex networks created by aggregation

We study aggregation as a mechanism for the creation of complex networks. In this evolution process vertices merge together, which increases a number of highly connected hubs. We study a range of complex network architectures produced by the aggregation. Fat-tailed (in particular, scale-free) distributions of connections are obtained for both networks with a finite number of vertices and growing networks. We observe a strong variation of a network structure with growing density of connections and find the phase transition of the condensation of edges. Finally, we demonstrate the importance of structural correlations in these networks.

Persistence properties of a system of coagulating and annihilating random walkers

We study a d-dimensional system of diffusing particles that on contact either annihilate with probability 1/(q-1) or coagulate with probability (q-2)/(q-1). In one dimension, the system models the zero-temperature Glauber dynamics of domain walls in the q-state Potts model. We calculate P(m,t), the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly (m-1) coagulations. Using perturbative renormalization group analysis for d<2, we show that, if the number of coagulations m is much less than the typical number M(t), then P(m,t) approximately m(zeta/d)t(-theta), with theta=dQ+Q(Q-1/2)epsilon+O(epsilon(2)), zeta=(2Q-1)epsilon+(2Q-1)(Q-1)(1/2+AQ)epsilon(2)+O(epsilon(3)), where Q=(q-1)/q, epsilon=2-d and A=-0.006....M(t) is shown to scale as M(t) approximately t(d/2-delta), where delta=d(1-Q)+(Q-1)(Q-1/2)epsilon+O(epsilon(2)). In two dimensions, we show that P(m,t) approximately ln(t)(Q(3-2Q))ln(m)((2Q-1)(2))t(-2Q) for m<<t(2Q-1). We also derive an exact nonperturbative relation between the exponents: namely delta(Q)=theta(1-Q). The one-dimensional results corresponding to epsilon=1 are compared with results from Monte Carlo simulations.

Cluster persistence in one-dimensional diffusion-limited cluster-cluster aggregation

The persistence probability, P(C)(t), of a cluster to remain unaggregated is studied in cluster-cluster aggregation, when the diffusion coefficient of a cluster depends on its size s as D(s) approximately s(gamma). In the mean field the problem maps to the survival of three annihilating random walkers with time-dependent noise correlations. For gamma> or =0 the motion of persistent clusters becomes asymptotically irrelevant and the mean-field theory provides a correct description. For gamma<0 the spatial fluctuations remain relevant and the persistence probability is overestimated by the random walk theory. The decay of persistence determines the small size tail of the cluster size distribution. For 0<gamma<2 the distribution is flat and, surprisingly, independent of gamma.