715 research outputs found

### 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

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