3,019 research outputs found
Two-Dimensional Scaling Limits via Marked Nonsimple Loops
We postulate the existence of a natural Poissonian marking of the double
(touching) points of SLE(6) and hence of the related continuum nonsimple loop
process that describes macroscopic cluster boundaries in 2D critical
percolation. We explain how these marked loops should yield continuum versions
of near-critical percolation, dynamical percolation, minimal spanning trees and
related plane filling curves, and invasion percolation. We show that this
yields for some of the continuum objects a conformal covariance property that
generalizes the conformal invariance of critical systems. It is an open problem
to rigorously construct the continuum objects and to prove that they are indeed
the scaling limits of the corresponding lattice objects.Comment: 25 pages, 5 figure
Invading interfaces and blocking surfaces in high dimensional disordered systems
We study the high-dimensional properties of an invading front in a disordered
medium with random pinning forces. We concentrate on interfaces described by
bounded slope models belonging to the quenched KPZ universality class. We find
a number of qualitative transitions in the behavior of the invasion process as
dimensionality increases. In low dimensions the system is characterized
by two different roughness exponents, the roughness of individual avalanches
and the overall interface roughness. We use the similarity of the dynamics of
an avalanche with the dynamics of invasion percolation to show that above
avalanches become flat and the invasion is well described as an annealed
process with correlated noise. In fact, for the overall roughness is
the same as the annealed roughness. In very large dimensions, strong
fluctuations begin to dominate the size distribution of avalanches, and this
phenomenon is studied on the Cayley tree, which serves as an infinite
dimensional limit. We present numerical simulations in which we measured the
values of the critical exponents of the depinning transition, both in finite
dimensional lattices with and on the Cayley tree, which support our
qualitative predictions. We find that the critical exponents in are very
close to their values on the Cayley tree, and we conjecture on this basis the
existence of a further dimension, where mean field behavior is obtained.Comment: 12 pages, REVTeX with 2 postscript figure
On the critical behavior of a lattice prey-predator model
The critical properties of a simple prey-predator model are revisited. For
some values of the control parameters, the model exhibits a line of directed
percolation like transitions to a single absorbing state. For other values of
the control parameters one finds a second line of continuous transitions toward
infinite number of absorbing states, and the corresponding steady-state
exponents are mean-field like. The critical behavior of the special point T
(bicritical point), where the two transition lines meet, belongs to a different
universality class. The use of dynamical Monte-Carlo method shows that a
particular strategy for preparing the initial state should be devised to
correctly describe the physics of the system near the second transition line.
Relationships with a forest fire model with immunization are also discussed.Comment: 6 RevTex pages, 7 ps figure
Cooperative Behavior of Kinetically Constrained Lattice Gas Models of Glassy Dynamics
Kinetically constrained lattice models of glasses introduced by Kob and
Andersen (KA) are analyzed. It is proved that only two behaviors are possible
on hypercubic lattices: either ergodicity at all densities or trivial
non-ergodicity, depending on the constraint parameter and the dimensionality.
But in the ergodic cases, the dynamics is shown to be intrinsically cooperative
at high densities giving rise to glassy dynamics as observed in simulations.
The cooperativity is characterized by two length scales whose behavior controls
finite-size effects: these are essential for interpreting simulations. In
contrast to hypercubic lattices, on Bethe lattices KA models undergo a
dynamical (jamming) phase transition at a critical density: this is
characterized by diverging time and length scales and a discontinuous jump in
the long-time limit of the density autocorrelation function. By analyzing
generalized Bethe lattices (with loops) that interpolate between hypercubic
lattices and standard Bethe lattices, the crossover between the dynamical
transition that exists on these lattices and its absence in the hypercubic
lattice limit is explored. Contact with earlier results are made via analysis
of the related Fredrickson-Andersen models, followed by brief discussions of
universality, of other approaches to glass transitions, and of some issues
relevant for experiments.Comment: 59 page
A topological approximation of the nonlinear Anderson model
We study the phenomena of Anderson localization in the presence of nonlinear
interaction on a lattice. A class of nonlinear Schrodinger models with
arbitrary power nonlinearity is analyzed. We conceive the various regimes of
behavior, depending on the topology of resonance-overlap in phase space,
ranging from a fully developed chaos involving Levy flights to pseudochaotic
dynamics at the onset of delocalization. It is demonstrated that quadratic
nonlinearity plays a dynamically very distinguished role in that it is the only
type of power nonlinearity permitting an abrupt localization-delocalization
transition with unlimited spreading already at the delocalization border. We
describe this localization-delocalization transition as a percolation
transition on a Cayley tree. It is found in vicinity of the criticality that
the spreading of the wave field is subdiffusive in the limit
t\rightarrow+\infty. The second moment grows with time as a powerlaw t^\alpha,
with \alpha = 1/3. Also we find for superquadratic nonlinearity that the analog
pseudochaotic regime at the edge of chaos is self-controlling in that it has
feedback on the topology of the structure on which the transport processes
concentrate. Then the system automatically (without tuning of parameters)
develops its percolation point. We classify this type of behavior in terms of
self-organized criticality dynamics in Hilbert space. For subquadratic
nonlinearities, the behavior is shown to be sensitive to details of definition
of the nonlinear term. A transport model is proposed based on modified
nonlinearity, using the idea of stripes propagating the wave process to large
distances. Theoretical investigations, presented here, are the basis for
consistency analysis of the different localization-delocalization patterns in
systems with many coupled degrees of freedom in association with the asymptotic
properties of the transport.Comment: 20 pages, 2 figures; improved text with revisions; accepted for
publication in Physical Review
The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on
a version of the triangular lattice in the complex plane have unique scaling
limits, which are invariant under rotations, scalings, and, in the case of the
MST, also under translations. However, they are not expected to be conformally
invariant. We also prove some geometric properties of the limiting MST. The
topology of convergence is the space of spanning trees introduced by Aizenman,
Burchard, Newman & Wilson (1999), and the proof relies on the existence and
conformal covariance of the scaling limit of the near-critical percolation
ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio
The scaling limits of near-critical and dynamical percolation
We prove that near-critical percolation and dynamical percolation on the
triangular lattice have a scaling limit as the mesh , in the "quad-crossing" space of percolation configurations
introduced by Schramm and Smirnov. The proof essentially proceeds by
"perturbing" the scaling limit of the critical model, using the pivotal
measures studied in our earlier paper. Markovianity and conformal covariance of
these new limiting objects are also established.Comment: 72 pages, 7 figures. Slightly revised, final versio
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