Lattice Animals Research Articles

On Monte Carlo generation and study of anisotropy of lattice animals

A new type of cluster in the universality class of lattice animals is introduced based on a stochastic form of the Martin algorithm by the Monte Carlo method. Lattice animals with sizes up to 1000 sites are generated. The asymptotic radius of the gyration exponent is consistent with the value 0.64, corresponding to a fractal dimension of 1.56. An investigation of anisotropy of these animals shows that small animals are anisotropic while larger ones are not.

Gelation and percolation : swelling effect

Small angle neutron scattering experiments were performed on Polyurethane branched polymers synthetized near the gelation threshold. The exponent τ of the size distribution function is measured, on a system cross-linked by polycondensation. We find τ = 2.2 ± 0.04. This leads to a fractal dimension for the polymers in the reaction bath Dp = 2.5 ± 0.09, in agreement with percolation model. In the dilute state, the fractal dimension is D = 1.98 ± 0.03, in agreement with recent predictions. Thus the largest branched polymers are swollen by dilution, and behave as lattice animals.

Possible relations for topological and transport properties of the lattice animal model of branched polymers

The author proposes that the following relations hold exactly for the lattice animal model of branched polymers. The resistivity exponent zeta a is exactly equal to da, the fractal dimensionality of the backbone of the anomals. The random walk fractal dimensionality dw is given by, dw=da+da, where da is the fractal dimensionality of the animals. The spectral dimension ds of the backbone of the animals is given by, ds=1 at all dimensions, i.e. the backbone is a highly decorated but essentially chain-like object. These are in contrast with earlier suggestions that one needs two (three) topological properties to obtain zeta a(dw). The author proposes that these relations may hold for all clusters for which loops are irrelevant and have a finite upper critical dimensionality. The implications of these results for diffusion-limited aggregates and percolation clusters are also discussed.

Polymer chain in an array of obstacles

A model “polymer chain in an array of obstacles”, which is more detailed and convenient variant of the generally accepted model “chain in a tube”, is proposed to be useful in molecular theories of systems with strongly entangled polymer chains. It is shown that the problem of the behaviour of the chain in the array of obstacles can be reduced to the problem of random walks on a tree (in the discrete realization) or on a Lobachevsky-Riemann space (in the continuous realization). The properties of the closed polymer chain in the array of obstacles are considered. The analogy between this problem and the problem of the statistics of lattice animals is established and it is shown that a long, closed, unentangled polymer chain in the array of obstacles takes a strongly collapsed conformation.

Directed self-avoiding walks on certain directed random fractals

Using the recently proposed real space renormalisation group method for directed systems, the authors study the critical behaviour of directed self-avoiding walks (DSAW) on both directed lattice animals (DLA) and directed percolation clusters at threshold pc (DPC) in two dimensions. The values for the exponent nu perpendicular to are found to be nu perpendicular to DSAWDLA approximately=0.590 and nu perpendicular to DSAWDPC approximately=0.528, both of which are higher than the mean-field value 0.5. It is also shown rigorously that nu /sub /// remains unchanged; i.e. nu /sub //DSAW/DLA= nu /sub //DSAW/DPC=1.

Topological and geometrical properties of random fractals

The underlying structure of random fractals is described in terms of two exponents, zeta and nu . The exponent zeta is sensitive only to the topology or 'connectedness' of the fractal, whereas nu depends only on the geometry of the fractal. The topological exponent arises from the scaling form for the distribution of paths, nl, on a finite fractal, where a path is defined as the shortest walk from one site to another. This path length distribution function is then used to derive expressions for the radius of gyration, pair correlation function, static structure factor and intensity of radiation scattered from percolation clusters at the gel point. From these calculations the fractal dimension is shown to be zeta / nu . Finally, recent numerical results for percolation clusters, percolation backbones, and lattice animals are discussed.

Recursion equations for static and kinetic models of branched polymers

A new renormalisation group treatment of bond lattice animals is presented which yields the critical bond fugacity and nu exponent for general dimension d. Good agreement is obtained with predictions based on different methods for all d. In particular nu =1- epsilon +O( epsilon 2), for d=1+ epsilon . Results are also given for a kinetic version of the bond lattice animal problem.

The backbone and conductivity of random clusters

The author studies the properties of the backbone of, and conduction in, random clusters which have homogeneous interior structure. These include the largest percolation cluster at pc, lattice animals and the Witten-Sander (WS) aggregates. For lattice animals the fractal dimension da of the backbone is estimated, the for first time, and is found to be about 1.14 at d=2 and about 1.39 at d=3. These values are much lower than the corresponding values for the animals themselves. The fractal and spectral dimensions of the backbone of the WS aggregates are estimated to be about 1.25 and 1.06 at d=2, respectively. A recent hypothesis which relates the fractal dimension of random walks on these clusters to those of the clusters and their backbones is also discussed. It is shown that for the WS aggregates ds=ds=2 on a Bethe lattice (i.e. at d= infinity ), in contrast with ds=4/3 and ds=1 for percolation clusters and lattice animals. The conductivity exponent ta of lattice animals is found to be ta(d=2) approximately=0.73 and ta(d=3) approximately=1.19 and ta(d>or=8)=2, whereas for the WS aggregates the author finds tWS(d=2) approximately=0.67 and tWS(d=3) approximately=0.94 and tWS(d to infinity )=1.

Lattice magnetic analog of branched polymers, lattice animals and percolation

Lattice magnetic analog of branched polymers, lattice animals and percolation

Corrections to scaling and phenomenological renormalization for 2-dimensional percolation and lattice animal problems

We continue and improve the transfer matrix approach of Derrida and de Seze by incorporating in two different ways the leading corrections to the asymptotic behaviour for wide strips. We find for the site percolation threshold in the square lattice p c =0.59274±0.00010, for the radius exponent of lattice animals 0.64075±0.00015, and for the inverse growth factor or critical fugacity 0.246150±0.000010 in the square lattice and 0.192925±0.000010 in the triangular lattice. These results are consistent with, and sometimes more accurate than, the best estimates published before Nous continuons et ameliorons l'approche par matrice de transfert de Derrida et de Seze en tenant compte de deux manieres differentes de la correction dominante. Nous obtenons pour le seuil de percolation de site sur le reseau carre p c =0,59274±0,00010, pour l'exposant qui caracterise la taille des animaux 0,64075±0,00015 et pour la fugacite critique 0,246150±0,000010 sur le reseau carre et 0,192925±0,000010 sur le reseau triangulaire. Ces resultats sont en accord, et parfois plus precis, que les meilleures estimations connues

A possible new exponent for directed lattice animals

The shape of directed lattice animals is studied by introducing a new exponent, νF, describing the divergence of the front width of the animal clusters. In two dimensions, the results from exact enumerations and real-space renormalization group method suggest that νF =0.43±0.03.

Comments on real-space renormalisation group for directed lattice animals

The authors carefully analyse the RG results of directed site lattice animals on a square lattice obtained from a new RSRG method proposed by them in 1984. Here, asymptotic forms for ν||(b + 1, b) and ν⊥(b + 1, b), the values obtained from cell-to-cell transformation of linear size b + 1 to b, are proposed in the large b limit. By fitting the data, they find ν|| = 0.796 and ν⊥ = 0.507. These results are in very good agreement with the known values.

Topological properties of diffusion limited aggregation and cluster-cluster aggregation

The detailed topological or 'connectivity' properties of the clusters formed in diffusion limited aggregation (DLA) and cluster-cluster aggregation (CCA) are considered for spatial dimensions d=2, 3 and 4. Specifically, for both aggregation phenomena the authors calculate the fractal dimension dmin= nu -1 defined by l approximately Rd(min) where l is the shortest path between two points separated by a Pythagorean distance R. For CCA, they find that dmin increases monotonically with d, presumably tending toward a limiting value dmin=2 at the upper critical dimensionality dc as found previously for lattice animals and percolation. For DLA, on the other hand, they find that dmin=1 within the accuracy of the calculations for d=2, 3 and 4, suggesting the absence of an upper critical dimension. They also discuss some of the subtle features encountered in calculating dmin for DLA.

Corrections to cluster-radius scaling for branched polymers and percolation

We report analyses of series enumerations for the mean radius of gyration for isotropic and directed lattice animals and for percolation clusters, in two and three dimensions. We allow for the leading correction to the scaling behaviour and obtain estimates of the leading correction-to-scaling exponent ϑ. We find ν-0.640±0.004 and ϑ=0.87±0.07 for isotropic animals in 2d, and ϑ=0.64±0.06 in 3d. For directed lattice animals we argue that the leading correction hasθ=ν⊥ orθ=ν⊥; we also estimateν∥=0.82±0.01 and 0.69 ±0.01 ind=2, 3 respectively. For percolation clusters at and abovepc, we find ϑ(pc) =0.58±0.06 and ϑ(p>pc)=0.84±0.09 in 2d, and ϑ(pc)=0.42±0.11 and ϑ(p>pc)=0.41 ±0.09 in 3d.

Caliper diameter of branched polymers

The authors report analyses of exact numerical data for the spanning diameter of two-dimensional lattice animals up to size N=17. Estimates of the exponent nu are consistent with previous studies. The leading correction to scaling has an exponent sigma approximately= nu which does not result from irrelevant variable effects. Interpretation of this correction as a 'surface' term is proposed.

Lattice statistics of branched polymers with specified topologies

The authors study the numbers of lattice animals with specified topologies. They prove that the growth constant for animals with c cycles and nk vertices of degree k (k=3, . . ., 2d), weakly embeddable in a d-dimensional hypercubic lattice, equal to mu , the growth constant for self-avoiding walks. For some particular topologies they derive upper and lower bounds for the corresponding critical exponents and estimate the values of these exponents by deriving and analysing new exact enumeration data. They conjecture that their previous result for trees with b branches (that the exponent is gamma +b-1, where gamma is the self-avoiding walk exponent) is also valid in the more general case in which the cyclomatic index (c) is non-zero; i.e. for b>or=1, the exponent does not depend on c. They show that their results are consistent with renormalisation group arguments that the universality class of branched polymers is independent of cycle and (non-zero) branching fugacities.

Directed site lattice animals with restricted valence

The critical behaviour of directed site lattice animals having valence no larger than v is studied in the square lattice for the cases of v=2 and 3 by using exact enumerations. The authors also study the cases where the microscopic restrictions are anisotropic so that the universality hypothesis can be tested for the systems possessing a non-trivial preferred axis. As expected, for both the isotropic and anisotropic restrictions, series analysis together with some exact results show that when v=3 the systems have the same critical behaviour as the unrestricted case while for v=2 the systems belong to the same universality class as directed self-avoiding walks.

On the relationship between the critical exponents of percolation conductivity and static exponents of percolation

The author argues that the critical exponent t of random conductance networks near the percolation threshold is given by t=(d-1) nu for low dimensionalities and t=1+ beta ' for high dimensionalities, where nu is the correlation length exponent, beta ' the backbone exponent and d is dimensionality. The author argues that what separates the two regimes is a critical fractal dimensionality Dl which equals 2. The author also argues that Dl is also a critical fractal dimensionality for fractals such as lattice animals and diffusion-limited aggregates. The result for low dimensionalities has been also obtained by Aharony and Stauffer by a different argument.

New method for growing branched polymers and large percolation clusters belowpc

We propose a new Monte Carlo method for generating large branched polymers; it is based on enrichment of percolation clusters below ${p}_{c}$. We find that one must take care to distinguish two different ensemble averages: one at constant mass $s$ and the other at constant chemical distance $l$. For constant $s$, we obtain clusters belonging to the universality class of lattice animals, while for constant $l$ we get topologically one-dimensional structures for all $d$.

Possible Breakdown of the Alexander-Orbach Rule at Low Dimensionalities

Simple conditions are presented under which the fractal dimension of a random walk on an aggregate, ${d}_{w}$, is given by ${d}_{w}=D+1$, where $D$ is the aggregate's fractal dimension. These conditions are argued (with one simple speculative assumption) to apply for $Dl2$, implying a breakdown of the Alexander-Orbach rule ${d}_{w}=\frac{3D}{2}$. Existing results for percolation clusters, lattice animals, and diffusion-limited aggregates seem to favor our new rule.