57,197 research outputs found
Random Growth Models
The link between a particular class of growth processes and random matrices
was established in the now famous 1999 article of Baik, Deift, and Johansson on
the length of the longest increasing subsequence of a random permutation.
During the past ten years, this connection has been worked out in detail and
led to an improved understanding of the large scale properties of
one-dimensional growth models. The reader will find a commented list of
references at the end. Our objective is to provide an introduction highlighting
random matrices. From the outset it should be emphasized that this connection
is fragile. Only certain aspects, and only for specific models, the growth
process can be reexpressed in terms of partition functions also appearing in
random matrix theory.Comment: Review paper; 24 pages, 4 figures; Minor correction
Dynamical Inequality in Growth Models
A recent exponent inequality is applied to a number of dynamical growth
models. Many of the known exponents for models such as the Kardar-Parisi-Zhang
(KPZ) equation are shown to be consistent with the inequality. In some cases,
such as the Molecular Beam Equation, the situation is more interesting, where
the exponents saturate the inequality. As the acid test for the relative
strength of four popular approximation schemes we apply the inequality to the
exponents obtained for two Non Local KPZ systems. We find that all methods but
one, the Self Consistent Expansion, violate the inequality in some regions of
parameter space. To further demonstrate the usefulness of the inequality, we
apply it to a specific model, which belongs to a family of models in which the
inequality becomes an equality. We thus show that the inequality can easily
yield results, which otherwise have to rely either on approximations or general
beliefs.Comment: 6 pages, 4 figure
Analysis of logistic growth models
A variety of growth curves have been developed to model both unpredated, intraspecific
population dynamics and more general biological growth. Most successful predictive models are
shown to be based on extended forms of the classical Verhulst logistic growth equation. We further
review and compare several such models and calculate and investigate properties of interest for
these. We also identify and detail several previously unreported associated limitations and restrictions.
A generalized form of the logistic growth curve is introduced which is shown incorporate these
models as special cases. The reported limitations of the generic growth model are shown to be addressed
by this new model and similarities between this and the extended growth curves are identified.
Several of its properties are also presented. We furthermore show that additional growth characteristics
are accommodated by this new model, enabling previously unsupported, untypical population dynamics to
be modelled by judicious choice of model parameter values alone
Random growth models with polygonal shapes
We consider discrete-time random perturbations of monotone cellular automata
(CA) in two dimensions. Under general conditions, we prove the existence of
half-space velocities, and then establish the validity of the Wulff
construction for asymptotic shapes arising from finite initial seeds. Such a
shape converges to the polygonal invariant shape of the corresponding
deterministic model as the perturbation decreases. In many cases, exact
stability is observed. That is, for small perturbations, the shapes of the
deterministic and random processes agree exactly. We give a complete
characterization of such cases, and show that they are prevalent among
threshold growth CA with box neighborhood. We also design a nontrivial family
of CA in which the shape is exactly computable for all values of its
probability parameter.Comment: Published at http://dx.doi.org/10.1214/009117905000000512 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
L\'{e}vy-based growth models
In the present paper, we give a condensed review, for the nonspecialist
reader, of a new modelling framework for spatio-temporal processes, based on
L\'{e}vy theory. We show the potential of the approach in stochastic geometry
and spatial statistics by studying L\'{e}vy-based growth modelling of planar
objects. The growth models considered are spatio-temporal stochastic processes
on the circle. As a by product, flexible new models for space--time covariance
functions on the circle are provided. An application of the L\'{e}vy-based
growth models to tumour growth is discussed.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6130 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
The Parallel Complexity of Growth Models
This paper investigates the parallel complexity of several non-equilibrium
growth models. Invasion percolation, Eden growth, ballistic deposition and
solid-on-solid growth are all seemingly highly sequential processes that yield
self-similar or self-affine random clusters. Nonetheless, we present fast
parallel randomized algorithms for generating these clusters. The running times
of the algorithms scale as , where is the system size, and the
number of processors required scale as a polynomial in . The algorithms are
based on fast parallel procedures for finding minimum weight paths; they
illuminate the close connection between growth models and self-avoiding paths
in random environments. In addition to their potential practical value, our
algorithms serve to classify these growth models as less complex than other
growth models, such as diffusion-limited aggregation, for which fast parallel
algorithms probably do not exist.Comment: 20 pages, latex, submitted to J. Stat. Phys., UNH-TR94-0
Growth models on the Bethe lattice
I report on an extensive numerical investigation of various discrete growth
models describing equilibrium and nonequilibrium interfaces on a substrate of a
finite Bethe lattice. An unusual logarithmic scaling behavior is observed for
the nonequilibrium models describing the scaling structure of the infinite
dimensional limit of the models in the Kardar-Parisi-Zhang (KPZ) class. This
gives rise to the classification of different growing processes on the Bethe
lattice in terms of logarithmic scaling exponents which depend on both the
model and the coordination number of the underlying lattice. The equilibrium
growth model also exhibits a logarithmic temporal scaling but with an ordinary
power law scaling behavior with respect to the appropriately defined lattice
size. The results may imply that no finite upper critical dimension exists for
the KPZ equation.Comment: 5 pages, 5 figure
Slow crack growth : models and experiments
The properties of slow crack growth in brittle materials are analyzed both
theoretically and experimentally. We propose a model based on a thermally
activated rupture process. Considering a 2D spring network submitted to an
external load and to thermal noise, we show that a preexisting crack in the
network may slowly grow because of stress fluctuations. An analytical solution
is found for the evolution of the crack length as a function of time, the time
to rupture and the statistics of the crack jumps. These theoretical predictions
are verified by studying experimentally the subcritical growth of a single
crack in thin sheets of paper. A good agreement between the theoretical
predictions and the experimental results is found. In particular, our model
suggests that the statistical stress fluctuations trigger rupture events at a
nanometric scale corresponding to the diameter of cellulose microfibrils.Comment: to be published in EPJ (European Physical Journal
On convergence in endogenous growth models
In this paper we analyze the rate of convergence to a balanced path in a class of endogenous growth models with physical and human capital. We show that such rate depends locally on the technological parameters of the model. but does not depend on those parameters related to preferences. These results stand in sharp contrast with those of the one-sector neoclassical growth model where both preferences and technologies determine the speed of convergence toward a steady state
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