559 research outputs found
Linear Relationship Statistics in Diffusion Limited Aggregation
We show that various surface parameters in two-dimensional diffusion limited
aggregation (DLA) grow linearly with the number of particles. We find the ratio
of the average length of the perimeter and the accessible perimeter of a DLA
cluster together with its external perimeters to the cluster size, and define a
microscopic schematic procedure for attachment of an incident new particle to
the cluster. We measure the fractal dimension of the red sites (i.e., the sites
upon cutting each of them splits the cluster) equal to that of the DLA cluster.
It is also shown that the average number of the dead sites and the average
number of the red sites have linear relationships with the cluster size.Comment: 4 pages, 5 figure
Droplets in the coexistence region of the two-dimensional Ising model
The two-dimensional Ising model with fixed magnetization is studied using
Monte Carlo techniques. At the coexistence line, the macroscopic, extensive
droplet of minority spins becomes thermally unstable by breaking up into
microscopic clusters. Intriguing finite--size effects as well as singularities
of thermal and cluster properties associated with the transition are discussed.Comment: 7 pages, 3 figures included, submitted to J. Phys. A: Math. Ge
Invaded cluster algorithm for a tricritical point in a diluted Potts model
The invaded cluster approach is extended to 2D Potts model with annealed
vacancies by using the random-cluster representation. Geometrical arguments are
used to propose the algorithm which converges to the tricritical point in the
two-dimensional parameter space spanned by temperature and the chemical
potential of vacancies. The tricritical point is identified as a simultaneous
onset of the percolation of a Fortuin-Kasteleyn cluster and of a percolation of
"geometrical disorder cluster". The location of the tricritical point and the
concentration of vacancies for q = 1, 2, 3 are found to be in good agreement
with the best known results. Scaling properties of the percolating scaling
cluster and related critical exponents are also presented.Comment: 8 pages, 5 figure
Percolation and cluster Monte Carlo dynamics for spin models
A general scheme for devising efficient cluster dynamics proposed in a
previous letter [Phys.Rev.Lett. 72, 1541 (1994)] is extensively discussed. In
particular the strong connection among equilibrium properties of clusters and
dynamic properties as the correlation time for magnetization is emphasized. The
general scheme is applied to a number of frustrated spin model and the results
discussed.Comment: 17 pages LaTeX + 16 figures; will appear in Phys. Rev.
Thermodynamics and statistical mechanics of frozen systems in inherent states
We discuss a Statistical Mechanics approach in the manner of Edwards to the
``inherent states'' (defined as the stable configurations in the potential
energy landscape) of glassy systems and granular materials. We show that at
stationarity the inherent states are distributed according a generalized Gibbs
measure obtained assuming the validity of the principle of maximum entropy,
under suitable constraints. In particular we consider three lattice models (a
diluted Spin Glass, a monodisperse hard-sphere system under gravity and a
hard-sphere binary mixture under gravity) undergoing a schematic ``tap
dynamics'', showing via Monte Carlo calculations that the time average of
macroscopic quantities over the tap dynamics and over such a generalized
distribution coincide. We also discuss about the general validity of this
approach to non thermal systems.Comment: 10 pages, 16 figure
Percolation transition and the onset of non exponential relaxation in fully frustrated models
We numerically study the dynamical properties of fully frustrated models in 2
and 3 dimensions. The results obtained support the hypothesis that the
percolation transition of the Kasteleyn-Fortuin clusters corresponds to the
onset of stretched exponential autocorrelation functions in systems without
disorder. This dynamical behavior may be due to the ``large scale'' effects of
frustration, present below the percolation threshold. Moreover these results
are consistent with the picture suggested by Campbell et al. in space of
configurations.Comment: 8 pages, 11 figures, revised versio
Thermodynamic versus Topological Phase Transitions: Cusp in the Kert\'esz Line
We present a study of phase transitions of the Curie--Weiss Potts model at
(inverse) temperature , in presence of an external field . Both
thermodynamic and topological aspects of these transitions are considered. For
the first aspect we complement previous results and give an explicit equation
of the thermodynamic transition line in the -- plane as well as the
magnitude of the jump of the magnetization (for . The signature
of the latter aspect is characterized here by the presence or not of a giant
component in the clusters of a Fortuin--Kasteleyn type representation of the
model. We give the equation of the Kert\'esz line separating (in the
-- plane) the two behaviours. As a result, we get that this line
exhibits, as soon as , a very interesting cusp where it
separates from the thermodynamic transition line
Non exponential relaxation in fully frustrated models
We study the dynamical properties of the fully frustrated Ising model. Due to
the absence of disorder the model, contrary to spin glass, does not exhibit any
Griffiths phase, which has been associated to non-exponential relaxation
dynamics. Nevertheless we find numerically that the model exhibits a stretched
exponential behavior below a temperature T_p corresponding to the percolation
transition of the Kasteleyn-Fortuin clusters. We have also found that the
critical behavior of this clusters for a fully frustrated q-state spin model at
the percolation threshold is strongly affected by frustration. In fact while in
absence of frustration the q=1 limit gives random percolation, in presence of
frustration the critical behavior is in the same universality class of the
ferromagnetic q=1/2-state Potts model.Comment: 7 pages, RevTeX, 11 figs, to appear on Physical Review
Beyond Blobs in Percolation Cluster Structure: The Distribution of 3-Blocks at the Percolation Threshold
The incipient infinite cluster appearing at the bond percolation threshold
can be decomposed into singly-connected ``links'' and multiply-connected
``blobs.'' Here we decompose blobs into objects known in graph theory as
3-blocks. A 3-block is a graph that cannot be separated into disconnected
subgraphs by cutting the graph at 2 or fewer vertices. Clusters, blobs, and
3-blocks are special cases of -blocks with , 2, and 3, respectively. We
study bond percolation clusters at the percolation threshold on 2-dimensional
square lattices and 3-dimensional cubic lattices and, using Monte-Carlo
simulations, determine the distribution of the sizes of the 3-blocks into which
the blobs are decomposed. We find that the 3-blocks have fractal dimension
in 2D and in 3D. These fractal dimensions are
significantly smaller than the fractal dimensions of the blobs, making possible
more efficient calculation of percolation properties. Additionally, the
closeness of the estimated values for in 2D and 3D is consistent with the
possibility that is dimension independent. Generalizing the concept of
the backbone, we introduce the concept of a ``-bone'', which is the set of
all points in a percolation system connected to disjoint terminal points
(or sets of disjoint terminal points) by disjoint paths. We argue that the
fractal dimension of a -bone is equal to the fractal dimension of
-blocks, allowing us to discuss the relation between the fractal dimension
of -blocks and recent work on path crossing probabilities.Comment: All but first 2 figs. are low resolution and are best viewed when
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