Aggregation Driven by a Localized Source

We study aggregation driven by a localized source of monomers. The densities become stationary and have algebraic tails far away from the source. We show that in a model with mass-independent reaction rates and diffusion coefficients, the density of monomers decays as $r^{-\beta(d)}$ in $d$ dimensions. The decay exponent has irrational values in physically relevant dimensions: $\beta(3)=(\sqrt{17}+1)/2$ and $\beta(2)=\sqrt{8}$. We also study Brownian coagulation with a localized source and establish the behavior of the total cluster density and the total number of of clusters in the system. The latter quantity exhibits a logarithmic growth with time.Comment: 9 page

Mass Exchange Processes with Input

We investigate a system of interacting clusters evolving through mass exchange and supplemented by input of small clusters. Three possibilities depending on the rate of exchange generically occur when input is homogeneous: continuous growth, gelation, and instantaneous gelation. We mostly study the growth regime using scaling methods. An exchange process with reaction rates equal to the product of reactant masses admits an exact solution which allows us to justify the validity of scaling approaches in this special case. We also investigate exchange processes with a localized input. We show that if the diffusion coefficients are mass-independent, the cluster mass distribution becomes stationary and develops an algebraic tail far away from the source.Comment: 14 pages, 2 fig

The power of choice in network growth

The "power of choice" has been shown to radically alter the behavior of a number of randomized algorithms. Here we explore the effects of choice on models of tree and network growth. In our models each new node has k randomly chosen contacts, where k > 1 is a constant. It then attaches to whichever one of these contacts is most desirable in some sense, such as its distance from the root or its degree. Even when the new node has just two choices, i.e., when k=2, the resulting network can be very different from a random graph or tree. For instance, if the new node attaches to the contact which is closest to the root of the tree, the distribution of depths changes from Poisson to a traveling wave solution. If the new node attaches to the contact with the smallest degree, the degree distribution is closer to uniform than in a random graph, so that with high probability there are no nodes in the network with degree greater than O(log log N). Finally, if the new node attaches to the contact with the largest degree, we find that the degree distribution is a power law with exponent -1 up to degrees roughly equal to k, with an exponential cutoff beyond that; thus, in this case, we need k >> 1 to see a power law over a wide range of degrees.Comment: 9 pages, 4 figure

Capture of the Lamb: Diffusing Predators Seeking a Diffusing Prey

We study the capture of a diffusing "lamb" by diffusing "lions" in one dimension. The capture dynamics is exactly soluble by probabilistic techniques when the number of lions is very small, and is tractable by extreme statistics considerations when the number of lions is very large. However, the exact solution for the general case of three or more lions is still not known.Comment: 11 pages, 6 figures, Invited paper for American Journal of Physic

Distinct Degrees and Their Distribution in Complex Networks

We investigate a variety of statistical properties associated with the number of distinct degrees that exist in a typical network for various classes of networks. For a single realization of a network with N nodes that is drawn from an ensemble in which the number of nodes of degree k has an algebraic tail, N_k ~ N/k^nu for k>>1, the number of distinct degrees grows as N^{1/nu}. Such an algebraic growth is also observed in scientific citation data. We also determine the N dependence of statistical quantities associated with the sparse, large-k range of the degree distribution, such as the location of the first hole (where N_k=0), the last doublet (two consecutive occupied degrees), triplet, dimer (N_k=2), trimer, etc.Comment: 12 pages, 6 figures, iop format. Version 2: minor correction

Lattice gases with a point source

We study diffusive lattice gases with local injection of particles, namely we assume that whenever the origin becomes empty, a new particle is immediately injected into the origin. We consider two lattice gases: a symmetric simple exclusion process and random walkers. The interplay between the injection events and the positions of the particles already present implies an effective collective interaction even for the ostensibly non-interacting random walkers. We determine the average total number of particles entering into the initially empty system. We also compute the average total number of distinct sites visited by all particles, and discuss the shape of the visited domain and the statistics of visits

Slowly Divergent Drift in the Field-Driven Lorentz Gas

The dynamics of a point charged particle which is driven by a uniform external electric field and moves in a medium of elastic scatterers is investigated. Using rudimentary approaches, we reproduce, in one dimension, the known results that the typical speed grows with time as t^{1/3} and that the leading behavior of the velocity distribution is exp(-|v|^3/t). In spatial dimension d>1, we develop an effective medium theory which provides a simple and comprehensive description for the motion of a test particle. This approach predicts that the typical speed grows as t^{1/3} for all d, while the speed distribution is given by the scaling form P(u,t)=^{-1}f(u/), where u=|v|^{3/2}, ~t^{1/2}, and f(z) is proportional to z^{(d-1)/3}exp(-z^2/2). For a periodic Lorentz gas with an infinite horizon, e. g., for a hypercubic lattice of scatters, a logarithmic correction to the effective medium result is predicted; in particular, the typical speed grows as (t ln t)^{1/3}.Comment: 10 pages, RevTeX, 5 ps figures include