Limiting Shapes of Ising Droplets, Ising Fingers, and Ising Solitons

We examine the evolution of an Ising ferromagnet endowed with zero-temperature single spin-flip dynamics. A large droplet of one phase in the sea of the opposite phase eventually disappears. An interesting behavior occurs in the intermediate regime when the droplet is still very large compared to the lattice spacing, but already very small compared to the initial size. In this regime the shape of the droplet is essentially deterministic (fluctuations are negligible in comparison with characteristic size). In two dimensions the shape is also universal, that is, independent on the initial shape. We analytically determine the limiting shape of the Ising droplet on the square lattice. When the initial state is a semi-infinite stripe of one phase in the sea of the opposite phase, it evolves into a finger which translates along its axis. We determine the limiting shape and the velocity of the Ising finger on the square lattice. An analog of the Ising finger on the cubic lattice is the translating Ising soliton. We show that far away from the tip, the cross-section of the Ising soliton coincides with the limiting shape of the two-dimensional Ising droplet and we determine a relation between the cross-section area, the distance from the tip, and the velocity of the soliton.Comment: 7 pages, 3 figures; version 2: two figures and references adde

Stochastic Dynamics of Growing Young Diagrams and Their Limit Shapes

We investigate a class of Young diagrams growing via the addition of unit cells and satisfying the constraint that the height difference between adjacent columns $\geq r$. In the long time limit, appropriately re-scaled Young diagrams approach a limit shape that we compute for each integer $r\geq 0$. We also determine limit shapes of `diffusively' growing Young diagrams satisfying the same constraint and evolving through the addition and removal of cells that proceed with equal rates.Comment: 18 pages, 5 figure

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

Kinetics of Deposition in the Diffusion-Controlled Limit

The adsorption of particles diffusing in a half-space bounded by the substrate and irreversibly sticking to the substrate upon contacts is investigated. We show that when absorbing particles are planar disks diffusing in the three-dimensional half-space, the coverage approaches its saturated jamming value as $t^{-1}$ in the large time limit [generally as $t^{-1/(d-1)}$ when the substrate is $d$ dimensional and $d>1$, and as $e^{-t/\ln(t)}$ when $d=1$]. We also analyze the asymptotic behavior when particles are spherical and when particles are planar aligned squares.Comment: 8 pages, 2 figure

Dynamical Critical Behaviors of the Ising Spin Chain: Swendsen-Wang and Wolff Algorithms

We study the zero-temperature Ising chain evolving according to the Swendsen-Wang dynamics. We determine analytically the domain length distribution and various ``historical'' characteristics, e.g., the density of unreacted domains is shown to scale with the average domain length as ^{-d} with d=3/2 (for the q-state Potts model, d=1+1/q). We also compute the domain length distribution for the Ising chain endowed with the zero-temperature Wolff dynamics.Comment: 12 pages, submitted to J. Phys.

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

Assortative Exchange Processes

In exchange processes clusters composed of elementary building blocks, monomers, undergo binary exchange in which a monomer is transferred from one cluster to another. In assortative exchange only clusters with comparable masses participate in exchange events. We study maximally assortative exchange processes in which only clusters of equal masses can exchange monomers. A mean-field framework based on rate equations is appropriate for spatially homogeneous systems in sufficiently high spatial dimension. For diffusion-controlled exchange processes, the mean-field approach is erroneous when the spatial dimension is smaller than critical; we analyze such systems using scaling and heuristic arguments. Apart from infinite-cluster systems we explore the fate of finite systems and study maximally assortative exchange processes driven by a localized input.Comment: 14 pages, 3 figure

Fixation in a cyclic Lotka-Volterra model

We study a cyclic Lotka-Volterra model of N interacting species populating a d-dimensional lattice. In the realm of a Kirkwood approximation, a critical number of species N_c(d) above which the system fixates is determined analytically. We find N_c=5,14,23 in dimensions d=1,2,3, in remarkably good agreement with simulation results in two dimensions.Comment: 4 pages, 2 figure

Phase Transition in a Traffic Model with Passing

We investigate a traffic model in which cars either move freely with quenched intrinsic velocities or belong to clusters formed behind slower cars. In each cluster, the next-to-leading car is allowed to pass and resume free motion. The model undergoes a phase transition from a disordered phase for the high passing rate to a jammed phase for the low rate. In the disordered phase, the cluster size distribution decays exponentially in the large size limit. In the jammed phase, the cluster size distribution has a power law tail and in addition there is an infinite-size cluster. Mean-field equations, describing the model in the framework of Maxwell approximation, correctly predict the existence of phase transition and adequately describe the disordered phase; properties of the jammed phase are studied numerically.Comment: 6 pages, 3 figure

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