62 research outputs found
Analytical Solution of a Stochastic Content Based Network Model
We define and completely solve a content-based directed network whose nodes
consist of random words and an adjacency rule involving perfect or approximate
matches, for an alphabet with an arbitrary number of letters. The analytic
expression for the out-degree distribution shows a crossover from a leading
power law behavior to a log-periodic regime bounded by a different power law
decay. The leading exponents in the two regions have a weak dependence on the
mean word length, and an even weaker dependence on the alphabet size. The
in-degree distribution, on the other hand, is much narrower and does not show
scaling behavior. The results might be of interest for understanding the
emergence of genomic interaction networks, which rely, to a large extent, on
mechanisms based on sequence matching, and exhibit similar global features to
those found here.Comment: 13 pages, 5 figures. Rewrote conclusions regarding the relevance to
gene regulation networks, fixed minor errors and replaced fig. 4. Main body
of paper (model and calculations) remains unchanged. Submitted for
publicatio
Log-periodic route to fractal functions
Log-periodic oscillations have been found to decorate the usual power law
behavior found to describe the approach to a critical point, when the
continuous scale-invariance symmetry is partially broken into a discrete-scale
invariance (DSI) symmetry. We classify the `Weierstrass-type'' solutions of the
renormalization group equation F(x)= g(x)+(1/m)F(g x) into two classes
characterized by the amplitudes A(n) of the power law series expansion. These
two classes are separated by a novel ``critical'' point. Growth processes
(DLA), rupture, earthquake and financial crashes seem to be characterized by
oscillatory or bounded regular microscopic functions g(x) that lead to a slow
power law decay of A(n), giving strong log-periodic amplitudes. In contrast,
the regular function g(x) of statistical physics models with
``ferromagnetic''-type interactions at equibrium involves unbound logarithms of
polynomials of the control variable that lead to a fast exponential decay of
A(n) giving weak log-periodic amplitudes and smoothed observables. These two
classes of behavior can be traced back to the existence or abscence of
``antiferromagnetic'' or ``dipolar''-type interactions which, when present,
make the Green functions non-monotonous oscillatory and favor spatial modulated
patterns.Comment: Latex document of 29 pages + 20 ps figures, addition of a new
demonstration of the source of strong log-periodicity and of a justification
of the general offered classification, update of reference lis
Growing Cayley trees described by Fermi distribution
We introduce a model for growing Cayley trees with thermal noise. The
evolution of these hierarchical networks reduces to the Eden model and the
Invasion Percolation model in the limit , respectively.
We show that the distribution of the bond strengths (energies) is described by
the Fermi statistics. We discuss the relation of the present results with the
scale-free networks described by Bose statistics
Generalized Dielectric Breakdown Model
We propose a generalized version of the Dielectric Breakdown Model (DBM) for
generic breakdown processes. It interpolates between the standard DBM and its
analog with quenched disorder, as a temperature like parameter is varied. The
physics of other well known fractal growth phenomena as Invasion Percolation
and the Eden model are also recovered for some particular parameter values. The
competition between different growing mechanisms leads to new non-trivial
effects and allows us to better describe real growth phenomena.
Detailed numerical and theoretical analysis are performed to study the
interplay between the elementary mechanisms. In particular, we observe a
continuously changing fractal dimension as temperature is varied, and report an
evidence of a novel phase transition at zero temperature in absence of an
external driving field; the temperature acts as a relevant parameter for the
``self-organized'' invasion percolation fixed point. This permits us to obtain
new insight into the connections between self-organization and standard phase
transitions.Comment: Submitted to PR
Laplacian Fractal Growth in Media with Quenched Disorder
We analyze the combined effect of a Laplacian field and quenched disorder for
the generation of fractal structures with a study, both numerical and
theoretical, of the quenched dielectric breakdown model (QDBM). The growth
dynamics is shown to evolve from the avalanches of invasion percolation (IP) to
the smooth growth of Laplacian fractals, i. e. diffusion limited aggregation
(DLA) and the dielectric breakdown model (DBM). The fractal dimension is
strongly reduced with respect to both DBM and IP, due to the combined effect of
memory and field screening. This implies a specific relation between the
fractal dimension of the breakdown structures (dielectric or mechanical) and
the microscopic properties of disordered materials.Comment: 11 pages Latex (revtex), 3 postscript figures included. Submitted to
PR
The Glassy Potts Model
We introduce a Potts model with quenched, frustrated disorder, that enjoys of
a gauge symmetry that forbids spontaneous magnetization, and allows the glassy
phase to extend from down to T=0. We study numerical the 4 dimensional
model with states. We show the existence of a glassy phase, and we
characterize it by studying the probability distributions of an order
parameter, the binder cumulant and the divergence of the overlap
susceptibility. We show that the dynamical behavior of the system is
characterized by aging.Comment: 4 pages including 4 (color) ps figures (all on page 4
Dynamics of Fractures in Quenched Disordered Media
We introduce a model for fractures in quenched disordered media. This model
has a deterministic extremal dynamics, driven by the energy function of a
network of springs (Born Hamiltonian). The breakdown is the result of the
cooperation between the external field and the quenched disorder. This model
can be considered as describing the low temperature limit for crack propagation
in solids. To describe the memory effects in this dynamics, and then to study
the resistance properties of the system we realized some numerical simulations
of the model. The model exhibits interesting geometric and dynamical
properties, with a strong reduction of the fractal dimension of the clusters
and of their backbone, with respect to the case in which thermal fluctuations
dominate. This result can be explained by a recently introduced theoretical
tool as a screening enhancement due to memory effects induced by the quenched
disorder.Comment: 7 pages, 9 Postscript figures, uses revtex psfig.sty, to be published
on Phys. Rev.
Glassy Dynamics of Protein Folding
A coarse grained model of a random polypeptide chain, with only discrete
torsional degrees of freedom and Hookean springs connecting pairs of
hydrophobic residues is shown to display stretched exponential relaxation under
Metropolis dynamics at low temperatures with the exponent , in
agreement with the best experimental results. The time dependent correlation
functions for fluctuations about the native state, computed in the Gaussian
approximation for real proteins, have also been found to have the same
functional form. Our results indicate that the energy landscape exhibits
universal features over a very large range of energies and is relatively
independent of the specific dynamics.Comment: RevTeX, 4 pages, multicolumn, including 5 figures; larger
computations performed, error bars improve
Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model
A fast method is presented for simulating the dielectric-breakdown model
using iterated conformal mappings. Numerical results for the dimension and for
corrections to scaling are in good agreement with the recent RG prediction of
an upper critical , at which a transition occurs between branching
fractal clusters and one-dimensional nonfractal clusters.Comment: 5 pages, 7 figures; corrections to scaling include
Non-perturbative renormalization of the KPZ growth dynamics
We introduce a non-perturbative renormalization approach which identifies
stable fixed points in any dimension for the Kardar-Parisi-Zhang dynamics of
rough surfaces. The usual limitations of real space methods to deal with
anisotropic (self-affine) scaling are overcome with an indirect functional
renormalization. The roughness exponent is computed for dimensions
to 8 and it results to be in very good agreement with the available
simulations. No evidence is found for an upper critical dimension. We discuss
how the present approach can be extended to other self-affine problems.Comment: 4 pages, 2 figures. To appear in Phys. Rev. Let
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