3,197 research outputs found
Interface growth in two dimensions: A Loewner-equation approach
The problem of Laplacian growth in two dimensions is considered within the
Loewner-equation framework. Initially the problem of fingered growth recently
discussed by Gubiec and Szymczak [T. Gubiec and P. Szymczak, Phys. Rev. E 77,
041602 (2008)] is revisited and a new exact solution for a three-finger
configuration is reported. Then a general class of growth models for an
interface growing in the upper-half plane is introduced and the corresponding
Loewner equation for the problem is derived. Several examples are given
including interfaces with one or more tips as well as multiple growing
interfaces. A generalization of our interface growth model in terms of
``Loewner domains,'' where the growth rule is specified by a time evolving
measure, is briefly discussed.Comment: To appear in Physical Review
Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile
The rotor-router model is a deterministic analogue of random walk. It can be
used to define a deterministic growth model analogous to internal DLA. We prove
that the asymptotic shape of this model is a Euclidean ball, in a sense which
is stronger than our earlier work. For the shape consisting of
sites, where is the volume of the unit ball in , we show that
the inradius of the set of occupied sites is at least , while the
outradius is at most for any . For a related
model, the divisible sandpile, we show that the domain of occupied sites is a
Euclidean ball with error in the radius a constant independent of the total
mass. For the classical abelian sandpile model in two dimensions, with particles, we show that the inradius is at least , and the
outradius is at most . This improves on bounds of Le Borgne
and Rossin. Similar bounds apply in higher dimensions.Comment: [v3] Added Theorem 4.1, which generalizes Theorem 1.4 for the abelian
sandpile. [v4] Added references and improved exposition in sections 2 and 4.
[v5] Final version, to appear in Potential Analysi
Quantitative estimates of discrete harmonic measures
A theorem of Bourgain states that the harmonic measure for a domain in
is supported on a set of Hausdorff dimension strictly less than
\cite{Bourgain}. We apply Bourgain's method to the discrete case, i.e., to the
distribution of the first entrance point of a random walk into a subset of , . By refining the argument, we prove that for all \b>0 there
exists \rho (d,\b)N(d,\b), any , and any | \{y\in\Z^d\colon \nu_{A,x}(y)
\geq n^{-\b} \}| \leq n^{\rho(d,\b)}, where denotes the
probability that is the first entrance point of the simple random walk
starting at into . Furthermore, must converge to as \b \to
\infty.Comment: 16 pages, 2 figures. Part (B) of the theorem is ne
Families of Vicious Walkers
We consider a generalisation of the vicious walker problem in which N random
walkers in R^d are grouped into p families. Using field-theoretic
renormalisation group methods we calculate the asymptotic behaviour of the
probability that no pairs of walkers from different families have met up to
time t. For d>2, this is constant, but for d<2 it decays as a power t^(-alpha),
which we compute to O(epsilon^2) in an expansion in epsilon=2-d. The second
order term depends on the ratios of the diffusivities of the different
families. In two dimensions, we find a logarithmic decay (ln t)^(-alpha'), and
compute alpha' exactly.Comment: 20 pages, 5 figures. v.2: minor additions and correction
Projected climate-induced faunal change in the western hemisphere
Climate change is predicted to be one of the greatest drivers of ecological change in the coming century. Increases in temperature over the last century have clearly been linked to shifts in species distributions. Given the magnitude of projected future climatic changes, we can expect even larger range shifts in the coming century. These changes will, in turn, alter ecological communities and the functioning of ecosystems. Despite the seriousness of predicted climate change, the uncertainty in climate-change projections makes it difficult for conservation managers and planners to proactively respond to climate stresses. To address one aspect of this uncertainty, we identified predictions of faunal change for which a high level of consensus was exhibited by different climate models. Specifically, we assessed the potential effects of 30 coupled atmosphereâocean general circulation model (AOGCM) future-climate simulations on the geographic ranges of 2954 species of birds, mammals, and amphibians in the Western Hemisphere. Eighty percent of the climate projections based on a relatively low greenhouse-gas emissions scenario result in the local loss of at least 10% of the vertebrate fauna over much of North and South America. The largest changes in fauna are predicted for the tundra, Central America, and the Andes Mountains where, assuming no dispersal constraints, specific areas are likely to experience over 90% turnover, so that faunal distributions in the future will bear little resemblance to those of today
Analysis of a fully packed loop model arising in a magnetic Coulomb phase
The Coulomb phase of spin ice, and indeed the Ic phase of water ice,
naturally realise a fully-packed two-colour loop model in three dimensions. We
present a detailed analysis of the statistics of these loops, which avoid
themselves and other loops of the same colour, and contrast their behaviour to
an analogous two-dimensional model. The properties of another extended degree
of freedom are also addressed, flux lines of the emergent gauge field of the
Coulomb phase, which appear as "Dirac strings" in spin ice. We mention
implications of these results for related models, and experiments.Comment: 5 pages, 4 figure
Determinantal Correlations of Brownian Paths in the Plane with Nonintersection Condition on their Loop-Erased Parts
As an image of the many-to-one map of loop-erasing operation \LE of random
walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk
(LERW) model is the statistical ensemble of SAWs such that the weight of each
SAW is given by the total weight of all random walks which are
inverse images of , \{\pi: \LE(\pi)=\zeta \}. We regard the Brownian
paths as the continuum limits of random walks and consider the statistical
ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the
LERW model. Following the theory of Fomin on nonintersecting LERWs, we
introduce a nonintersecting system of -tuples of LEBPs in a domain in
the complex plane, where the total weight of nonintersecting LEBPs is given by
Fomin's determinant of an matrix whose entries are boundary
Poisson kernels in . We set a sequence of chambers in a planar domain and
observe the first passage points at which Brownian paths first enter each chamber, under the condition that the loop-erased
parts (\LE(\gamma_1),..., \LE(\gamma_N)) make a system of nonintersecting
LEBPs in the domain in the sense of Fomin. We prove that the correlation
functions of first passage points of the Brownian paths of the present system
are generally given by determinants specified by a continuous function called
the correlation kernel. The correlation kernel is of Eynard-Mehta type, which
has appeared in two-matrix models and time-dependent matrix models studied in
random matrix theory. Conformal covariance of correlation functions is
demonstrated.Comment: v3: REVTeX4, 27 pages, 10 figures, corrections made for publication
in Phys.Rev.
The Hamburg/ESO R-process Enhanced Star survey (HERES). V. Detailed abundance analysis of the r-process enhanced star HE 2327-5642
We report on a detailed abundance analysis of the strongly r-process enhanced
giant star, HE 2327-5642 ([Fe/H] = -2.78, [r/Fe] = +0.99). Determination of
stellar parameters and element abundances was based on analysis of high-quality
VLT/UVES spectra. The surface gravity was calculated from the NLTE ionization
balance between Fe I and Fe II, and Ca I and Ca II. Accurate abundances for a
total of 40 elements and for 23 neutron-capture elements beyond Sr and up to Th
were determined. The heavy element abundance pattern of HE 2327-5642 is in
excellent agreement with those previously derived for other strongly r-process
enhanced stars. Elements in the range from Ba to Hf match the scaled Solar
r-process pattern very well. No firm conclusion can be drawn with respect to a
relationship between the fisrt neutron-capture peak elements, Sr to Pd, in HE
2327-5642 and the Solar r-process, due to the uncertainty of the latter. A
clear distinction in Sr/Eu abundance ratios was found between the halo stars
with different europium enhancement. The strongly r-process enhanced stars
reveal a low Sr/Eu abundance ratio at [Sr/Eu] = -0.92+-0.13, while the stars
with 0 < [Eu/Fe] < 1 and [Eu/Fe] < 0 have 0.36 dex and 0.93 dex larger Sr/Eu
values, respectively. Radioactive dating for HE 2327-5642 with the observed
thorium and rare-earth element abundance pairs results in an average age of
13.3 Gyr, when based on the high-entropy wind calculations, and 5.9 Gyr, when
using the Solar r-residuals. HE 2327-5642 is suspected to be radial-velocity
variable based on our high-resolution spectra, covering ~4.3 years.Comment: 16 pages, 12 figures, accepted to A&
On harmonic measure of critical curves
Fractal geometry of critical curves appearing in 2D critical systems is
characterized by their harmonic measure. For systems described by conformal
field theories with central charge , scaling exponents of
harmonic measure have been computed by B. Duplantier [Phys. Rev. Lett. {\bf
84}, 1363 (2000)] by relating the problem to boundary two-dimensional gravity.
We present a simple argument that allows us to connect harmonic measure of
critical curves to operators obtained by fusion of primary fields, and compute
characteristics of fractal geometry by means of regular methods of conformal
field theory. The method is not limited to theories with .Comment: Some more correction
A Model Ground State of Polyampholytes
The ground state of randomly charged polyampholytes is conjectured to have a
structure similar to a necklace, made of weakly charged parts of the chain,
compacting into globules, connected by highly charged stretched `strings'. We
suggest a specific structure, within the necklace model, where all the neutral
parts of the chain compact into globules: The longest neutral segment compacts
into a globule; in the remaining part of the chain, the longest neutral segment
(the 2nd longest neutral segment) compacts into a globule, then the 3rd, and so
on. We investigate the size distributions of the longest neutral segments in
random charge sequences, using analytical and Monte Carlo methods. We show that
the length of the n-th longest neutral segment in a sequence of N monomers is
proportional to N/(n^2), while the mean number of neutral segments increases as
sqrt(N). The polyampholyte in the ground state within our model is found to
have an average linear size proportional to sqrt(N), and an average surface
area proportional to N^(2/3).Comment: 8 two-column pages. 5 eps figures. RevTex. Submitted to Phys. Rev.
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