19,352 research outputs found
Statistics of lattice animals (polyominoes) and polygons
We have developed an improved algorithm that allows us to enumerate the
number of site animals (polyominoes) on the square lattice up to size 46.
Analysis of the resulting series yields an improved estimate, , for the growth constant of lattice animals and confirms to a very
high degree of certainty that the generating function has a logarithmic
divergence. We prove the bound We also calculate the radius
of gyration of both lattice animals and polygons enumerated by area. The
analysis of the radius of gyration series yields the estimate , for both animals and polygons enumerated by area. The mean
perimeter of polygons of area is also calculated. A number of new amplitude
estimates are given.Comment: 10 pages, 2 eps figure
Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals
We consider nearest-neighbor self-avoiding walk, bond percolation, lattice
trees, and bond lattice animals on . The two-point functions of
these models are respectively the generating function for self-avoiding walks
from the origin to , the probability of a connection from
the origin to , and the generating functions for lattice trees or lattice
animals containing the origin and . Using the lace expansion, we prove that
the two-point function at the critical point is asymptotic to
as , for for self-avoiding
walk, for for percolation, and for sufficiently large for lattice
trees and animals. These results are complementary to those of [Ann. Probab. 31
(2003) 349--408], where spread-out models were considered. In the course of the
proof, we also provide a sufficient (and rather sharp if ) condition under
which the two-point function of a random walk on is
asymptotic to as .Comment: Published in at http://dx.doi.org/10.1214/009117907000000231 the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Connection Between Percolation and Lattice Animals
An n-state Potts lattice gas Hamiltonian is constructed whose partition function is shown to reproduce in the limit n→0 the generating function for the statistics of either lattice animals or percolating clusters for appropriate choices of potentials. This model treats an ensemble of single clusters terminated by weighted perimeter bonds rather than clusters distributed uniformly throughout the lattice. The model is studied within mean-field theory as well as via the ε expansion. In general, cluster statistics are described by the lattice animal\u27s fixed point. The percolation fixed point appears as a multicritical point in a space of potentials not obviously related to that of the usual one-state Potts model
Generalized Percolation
A generalized model of percolation encompassing both the usual model, in which bonds are occupied with probability p and are vacant with probability (1−p), and the model appropriate to the statistics of lattice animals, in which the fugacity for occupied bonds is p and that for unoccupied bonds is unity, is formulated. Within this model we discuss the crossover between the two problems and we study the statistics of large clusters. We determine the scaling form which the distribution function for the number of clusters with a given number of sites n assumes as a function of both n and p. For p near pc we find that the distribution function depends on percolation exponents for u=n(pc−p)Δp small, where Δp is a crossover exponent, and on exponents appropriate to the lattice-animals problem for large values of u. We thus have displayed the relation between the two limits and show conclusively that the lattice-animals exponents cannot be obtained by any simple scaling arguments from the percolation exponents. We also demonstrate that essential singularities in the cluster distribution functions for p\u3epc arise from metastable states of the Potts model
Simulations of lattice animals and trees
The scaling behaviour of randomly branched polymers in a good solvent is
studied in two to nine dimensions, using as microscopic models lattice animals
and lattice trees on simple hypercubic lattices. As a stochastic sampling
method we use a biased sequential sampling algorithm with re-sampling, similar
to the pruned-enriched Rosenbluth method (PERM) used extensively for linear
polymers. Essentially we start simulating percolation clusters (either site or
bond), re-weigh them according to the animal (tree) ensemble, and prune or
branch the further growth according to a heuristic fitness function. In
contrast to previous applications of PERM, this fitness function is {\it not}
the weight with which the actual configuration would contribute to the
partition sum, but is closely related to it. We obtain high statistics of
animals with up to several thousand sites in all dimension 2 <= d <= 9. In
addition to the partition sum (number of different animals) we estimate
gyration radii and numbers of perimeter sites. In all dimensions we verify the
Parisi-Sourlas prediction, and we verify all exactly known critical exponents
in dimensions 2, 3, 4, and >= 8. In addition, we present the hitherto most
precise estimates for growth constants in d >= 3. For clusters with one site
attached to an attractive surface, we verify the superuniversality of the
cross-over exponent at the adsorption transition predicted by Janssen and
Lyssy. Finally, we discuss the collapse of animals and trees, arguing that our
present version of the algorithm is also efficient for some of the models
studied in this context, but showing that it is {\it not} very efficient for
the `classical' model for collapsing animals.Comment: 17 pages RevTeX, 29 figures include
1/σ Expansion for Quantum Percolation
A method for obtaining a 1/σ expansion for certain statistical models is presented, where σ+1 is the coordination number of the lattice. The method depends on being able to generate exact recursion relations for the Cayley tree. By perturbing the recursion relation to take account of the dominant loops in a hypercubic lattice for large σ, we obtain corrections of order σ−2 to the recursion relations. For the tight-binding model on random bond clusters (the quantum-percolation problem) we obtain corrections to this order for the critical concentration p* at which the transition between localized and extended zero-energy eigenfunctions takes place. It is believed that this concentration coincides with the transition when all energies are considered. In addition, we display the relation for the Cayley tree between quantum-percolation and lattice animals (or dilute branched polymers). We show that this relation manifests itself in the appearance of singularities in the quantum-percolation problem at negative concentration which correspond to the physical transition at positive fugacity in the statistics of lattice animals. Corrections of order σ−2 to the location of this unphysical singularity in the quantum-percolation problem are also obtained
Lattice Statistics Of Polymer Adsorption
The interaction of branched polymers with an adsorption surface is studied using rigorous and numerical methods. For a polymer network with a fixed topology and consisting of self-avoiding chains, we prove that the reduced free energy is the same as that for self-avoiding walks interacting with a surface. For a network modelled by a lattice animal, we prove that a phase transition exists when such an animal interacts with a surface. The transition points are numerically studied by one and two variable Pade approximants. A number of rigorous results for the statistics of lattice animals are also obtained
Violating conformal invariance: Two-dimensional clusters grafted to wedges, cones, and branch points of Riemann surfaces
We present simulations of 2-d site animals on square and triangular lattices
in non-trivial geomeLattice animals are one of the few critical models in
statistical mechanics violating conformal invariance. We present here
simulations of 2-d site animals on square and triangular lattices in
non-trivial geometries. The simulations are done with the newly developed PERM
algorithm which gives very precise estimates of the partition sum, yielding
precise values for the entropic exponent (). In particular, we studied animals grafted to the tips of wedges
with a wide range of angles , to the tips of cones (wedges with the
sides glued together), and to branching points of Riemann surfaces. The latter
can either have sheets and no boundary, generalizing in this way cones to
angles degrees, or can have boundaries, generalizing wedges. We
find conformal invariance behavior, , only for small
angles (), while for
. These scalings hold both for wedges and cones. A heuristic
(non-conformal) argument for the behavior at large is given, and
comparison is made with critical percolation.Comment: 4 pages, includes 3 figure
A framework and simulation engine for studying artificial life
The area of computer-generated artificial life-forms is a relatively recent
field of inter-disciplinary study that involves mathematical modelling, physical
intuition and ideas from chemistry and biology and computational science.
Although the attribution of “life” to non biological systems is still controversial,
several groups agree that certain emergent properties can be ascribed to
computer simulated systems that can be constructed to “live” in a simulated
environment. In this paper we discuss some of the issues and infrastructure
necessary to construct a simulation laboratory for the study of computer generated
artificial life-forms. We review possible technologies and present some
preliminary studies based around simple models
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