1,712 research outputs found
Two-dimensional volume-frozen percolation: exceptional scales
We study a percolation model on the square lattice, where clusters "freeze"
(stop growing) as soon as their volume (i.e. the number of sites they contain)
gets larger than N, the parameter of the model. A model where clusters freeze
when they reach diameter at least N was studied in earlier papers. Using volume
as a way to measure the size of a cluster - instead of diameter - leads, for
large N, to a quite different behavior (contrary to what happens on the binary
tree, where the volume model and the diameter model are "asymptotically the
same"). In particular, we show the existence of a sequence of "exceptional"
length scales.Comment: 20 pages, 2 figure
Jamming percolation and glassy dynamics
We present a detailed physical analysis of the dynamical glass-jamming
transition which occurs for the so called Knight models recently introduced and
analyzed in a joint work with D.S.Fisher \cite{letterTBF}. Furthermore, we
review some of our previous works on Kinetically Constrained Models.
The Knights models correspond to a new class of kinetically constrained
models which provide the first example of finite dimensional models with an
ideal glass-jamming transition. This is due to the underlying percolation
transition of particles which are mutually blocked by the constraints. This
jamming percolation has unconventional features: it is discontinuous (i.e. the
percolating cluster is compact at the transition) and the typical size of the
clusters diverges faster than any power law when . These
properties give rise for Knight models to an ergodicity breaking transition at
: at and above a finite fraction of the system is frozen. In
turn, this finite jump in the density of frozen sites leads to a two step
relaxation for dynamic correlations in the unjammed phase, analogous to that of
glass forming liquids. Also, due to the faster than power law divergence of the
dynamical correlation length, relaxation times diverge in a way similar to the
Vogel-Fulcher law.Comment: Submitted to the special issue of Journal of Statistical Physics on
Spin glasses and related topic
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
Critical percolation in the dynamics of the 2d ferromagnetic Ising model
We study the early time dynamics of the 2d ferromagnetic Ising model
instantaneously quenched from the disordered to the ordered, low temperature,
phase. We evolve the system with kinetic Monte Carlo rules that do not conserve
the order parameter. We confirm the rapid approach to random critical
percolation in a time-scale that diverges with the system size but is much
shorter than the equilibration time. We study the scaling properties of the
evolution towards critical percolation and we identify an associated growing
length, different from the curvature driven one. By working with the model
defined on square, triangular and honeycomb microscopic geometries we establish
the dependence of this growing length on the lattice coordination. We discuss
the interplay with the usual coarsening mechanism and the eventual fall into
and escape from metastability.Comment: 67 pages, 33 figure
A percolation process on the binary tree where large finite clusters are frozen
We study a percolation process on the planted binary tree, where clusters
freeze as soon as they become larger than some fixed parameter N. We show that
as N goes to infinity, the process converges in some sense to the frozen
percolation process introduced by Aldous. In particular, our results show that
the asymptotic behaviour differs substantially from that on the square lattice,
on which a similar process has been studied recently by van den Berg, de Lima
and Nolin.Comment: 11 page
Percolation with constant freezing
We introduce and study a model of percolation with constant freezing (PCF)
where edges open at constant rate 1, and clusters freeze at rate \alpha
independently of their size. Our main result is that the infinite volume
process can be constructed on any amenable vertex transitive graph. This is in
sharp contrast to models of percolation with freezing previously introduced,
where the limit is known not to exist. Our interest is in the study of the
percolative properties of the final configuration as a function of \alpha. We
also obtain more precise results in the case of trees. Surprisingly the
algebraic exponent for the cluster size depends on the degree, suggesting that
there is no lower critical dimension for the model. Moreover, even for
\alpha<\alpha_c, it is shown that finite clusters have algebraic tail decay,
which is a signature of self organised criticality. Partial results are
obtained on Z^d, and many open questions are discussed.Comment: 30 pages, 8 figure
Glassy behavior of the site frustrated percolation model
The dynamical properties of the site frustrated percolation model are
investigated and compared with those of glass forming liquids. When the density
of the particles on the lattice becomes high enough, the dynamics of the model
becomes very slow, due to geometrical constraints, and rearrangement on large
scales is needed to allow relaxation. The autocorrelation functions, the
specific volume for different cooling rates, and the mean square displacement
are evaluated, and are found to exhibit glassy behavior.Comment: 8 pages, RevTeX, 11 fig
Generalized contact process on random environments
Spreading from a seed is studied by Monte Carlo simulation on a square
lattice with two types of sites affecting the rates of birth and death. These
systems exhibit a critical transition between survival and extinction. For
time- dependent background, this transition is equivalent to those found in
homogeneous systems (i.e. to directed percolation). For frozen backgrounds, the
appearance of Griffiths phase prevents the accurate analysis of this
transition. For long times in the subcritical region, spreading remains
localized in compact (rather than ramified) patches, and the average number of
occupied sites increases logarithmically in the surviving trials.Comment: 6 pages, 7 figure
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