Lattice Animals Research Articles

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.

Critical Exponents in Percolation via Lattice Animals

We examine the percolation model on $\mathbb{Z}^d$ by an approach involving lattice animals and their surface-area-to-volume ratio. For $\beta \in [0,2(d-1))$, let $f(\beta)$ be the asymptotic exponential rate in the number of edges of the number of lattice animals containing the origin which have surface-area-to-volume ratio $\beta$. The function $f$ is bounded above by a function which may be written in an explicit form. For low values of $\beta$ ($\beta \leq 1/p_c - 1$), equality holds, as originally demonstrated by F. Delyon. For higher values ($\beta > 1/p_c - 1$), the inequality is strict. We introduce two critical exponents, one of which describes how quickly $f$ falls away from the explicit form as $\beta$ rises from $1/p_c - 1$, and the second of which describes how large clusters appear in the marginally subcritical regime of the percolation model. We demonstrate that the pair of exponents must satisfy certain inequalities. Other such inequalities yield sufficient conditions for the absence of an infinite cluster at the critical value (c.f. [4]). The first exponent is related to one of a more conventional nature in the scaling theory of percolation, that of correlation size. In deriving this relation, we find that there are two possible behaviours, depending on the value of the first exponent, for the typical surface-area-to-volume ratio of an unusually large cluster in the marginally subcritical regime.

Off-Lattice Simulation of Multifunctional Monomer Polymerizations: Effects of Monomer Mobility, Structure, and Functionality on Structural Evolution at Low Conversion

Fundamental insight into the polymerization behavior and structural evolution of highly cross-linked polymers synthesized from the chain polymerization of multifunctional monomers will benefit a number of current and emerging applications. This contribution reintroduces1 an off-lattice kinetic gelation model with refinements to the representations of monomer mobility, structure, and functionality, which more realistically capture low conversion propagation events during cross-linked network evolution. Intramolecularly cross-linked macromolecules (ICMs) are formed by simulation of a single propagating radical within a volume containing ca. 100 000 monomer units. Effects of monomer mobility, structure, and functionality on the relative size, kinetic chain length, and intramolecular cross-link density of ICMs are reported. Finally, aspects of percolation theory are applied to this off-lattice kinetic gelation simulation. The fractal dimension corresponds to the theoretical value for a lattice animal created by a three-dimensional percolation model and it is independent of conversion. Overall, this contribution demonstrates the utility of a unique approach for kinetic gelation simulation of multifunctional monomer polymerizations that affords more realistic representation of monomer structure and dynamics than traditional, lattice-based models.

Local excitations of a spin glass in a magnetic field

We study the minimum energy clusters (MEC) above the ground state for the 3-d Edwards-Anderson Ising spin glass in a magnetic field. For fields B below 0.4, we find that the field has almost no effect on the excitations that we can probe, of volume V <= 64. As found previously for B=0, their energies decrease with V, and their magnetization remains very small (even slightly negative). For larger fields, both the MEC energy and magnetization grow with V, as expected in a paramagnetic phase. However, all results appear to scale as BV (instead of the B sqrt(V) expected from droplet arguments), suggesting that the spin glass phase is destroyed by any small field. Finally, the geometry of the MEC is completely insensitive to the field, giving further credence that they are lattice animals, in the presence or the absence of a field.

Statistical mechanics of directed models of polymers in the square lattice

Directed square lattice models of polymers and vesicles have received considerable attention in the recent mathematical and physical sciences literature. These are idealized geometric directed lattice models introduced to study phase behaviour in polymers, and include Dyck paths, partially directed paths, directed trees and directed vesicles models. Directed models are closely related to models studied in the combinatorics literature (and are often exactly solvable). They are also simplified versions of a number of statistical mechanics models, including the self-avoiding walk, lattice animals and lattice vesicles. The exchange of approaches and ideas between statistical mechanics and combinatorics have considerably advanced the description and understanding of directed lattice models, and this will be explored in this review. The combinatorial nature of directed lattice path models makes a study using generating function approaches most natural. In contrast, the statistical mechanics approach would introduce partition functions and free energies, and then investigate these using the general framework of critical phenomena. Generating function and statistical mechanics approaches are closely related. For example, questions regarding the limiting free energy may be approached by considering the radius of convergence of a generating function, and the scaling properties of thermodynamic quantities are related to the asymptotic properties of the generating function. In this review the methods for obtaining generating functions and determining free energies in directed lattice path models of linear polymers is presented. These methods include decomposition methods leading to functional recursions, as well as the Temperley method (that is implemented by creating a combinatorial object, one slice at a time). A constant term formulation of the generating function will also be reviewed. The thermodynamic features and critical behaviour in models of directed paths may be informative about the underlying properties that determine phase diagrams for wider classes of models, including physical models of polymers. Of particular interest are adsorption and collapse transitions in models of polymers and copolymers. The properties of thermodynamic quantities in those models are described by tricritical scaling. This is reviewed for directed path models, and the generating function approaches can be used to apply tricritical scaling to models of adsorbing, inflating and collapsing directed lattice paths. Critical exponents for a variety of models can be obtained in this manner, and with it a better understanding, and a classification, of the models.

Some problems in the counting of lattice animals, polyominoes, polygons and walks

Some problems in the counting of lattice animals, polyominoes, polygons and walks

Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models

We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ${\mathbb{Z}^d}$, having long finite-range connections, above their upper critical dimensions $d=4$ (self-avoiding walk), $d=6$ (percolation) and $d=8$ (trees and animals). The two-point functions for these models are respectively the generating function for self-avoiding walks from the origin to $x \in {\mathbb{Z}^d}$, the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of $|x|^{2-d}$ as $x \to \infty$. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoiding walk, the triangle condition for percolation, and the square condition for lattice trees and lattice animals, for sufficiently spread-out models above the upper critical dimension.

Lattice animals and heaps of dimers

The general quest of this paper is the search for new classes of square lattice animals that are both large and exactly enumerable. The starting point is a bijection between a subclass of animals, called directed animals, and certain heaps of dimers, called pyramids, which was described by Viennot more than 10 years ago. The generating function for directed animals had been known since 1982, but Viennot's bijection suggested a new approach that greatly simplified its derivation. We define here two natural classes of heaps that are supersets of pyramids and are in bijection with certain classes of animals, and we enumerate them exactly. The first class has an algebraic generating function and growth constant 3.5 (meaning that the number of n-celled animals grows like 3.5 n ), while the other has a transcendental non-holonomic generating function and growth constant 3.58… . The generating function for directed animals is algebraic, and has growth constant 3. Hence both these new classes are exponentially larger. We obtain similar results for triangular lattice animals.

On directed interacting animals and directed percolation

We study the phase diagram of fully directed lattice animals with nearest-neighbour interactions on the square lattice. This model comprises several interesting ensembles (directed site and bond trees, bond animals, strongly embeddable animals) as special cases and its collapse transition is equivalent to a directed bond percolation threshold. Precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n=17. The crossover region in the vicinity of the collapse transition is analyzed in detail and the crossover exponent $\phi$ is determined directly from the singular part of the free energy. We show using scaling arguments and an exact relation due to Dhar that $\phi$ is equal to the Fisher exponent $\sigma$ governing the size distribution of large directed percolation clusters.

Critical and multicritical semi-random (1 +d)-dimensional lattices and hard objects inddimensions

We investigate models of (1 + d)D Lorentzian semi-random lattices with one random (space-like) direction and d regular (time-like) ones. We prove a general inversion formula expressing the partition function of these models as the inverse of that of hard objects in d dimensions. This allows for an exact solution of a variety of new models including critical and multicritical generalized (1+1)D Lorentzian surfaces, with fractal dimensions dF = k + 1, k = 1,2,3,... , as well as a new model of (1+2)D critical tetrahedral complexes, with fractal dimension dF = 12/5. Critical exponents and universal scaling functions follow from this solution. We finally establish a general connection between (1 + d)D Lorentzian lattices and directed-site lattice animals in (1 + d) dimensions.

Diffusion in disordered media

Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the 'ant in the labyrinth') on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures.

Linear growth for greedy lattice animals

Let d⩾2, and let {X v , v ∈ Z d} be an i.i.d. family of non-negative random variables with common distribution F. Let N( n) be the maximum value of ∑ v∈ ξ X v over all connected subsets ξ of Z d of size n which contain the origin. This model of “greedy lattice animals” was introduced by Cox et al. (Ann. Appl. Probab. 3 (1993) 1151) and Gandolfi and Kesten (Ann. Appl. Probab. 4 (1994) 76), who showed that if EX 0 d( log + X 0 ) d+ε<∞ for some ε>0, then N( n)/ n→ N a.s. and in L 1 for some N<∞. Using related but partly simpler methods, we derive the same conclusion under the slightly weaker condition that ∫ 0 ∞ (1−F(x)) 1/d dx<∞ , and show that N⩽c ∫ 0 ∞ (1−F(x)) 1/d dx for some constant c. We also give analogous results for the related “greedy lattice paths” model.

Potts model partition functions for self-dual families of strip graphs

We consider the q-state Potts model on families of self-dual strip graphs G D of the square lattice of width L y and arbitrarily great length L x , with periodic longitudinal boundary conditions. The general partition function Z and the T=0 antiferromagnetic special case P (chromatic polynomial) have the respective forms ∑ j=1 N F, L y , λ c F, L y , j ( λ F, L y , j ) L x , with F= Z, P. For arbitrary L y , we determine (i) the general coefficient c F, L y , j in terms of Chebyshev polynomials, (ii) the number n F ( L y , d) of terms with each type of coefficient, and (iii) the total number of terms N F, L y , λ . We point out interesting connections between the n Z ( L y , d) and Temperley–Lieb algebras, and between the N F, L y , λ and enumerations of directed lattice animals. Exact calculations of P are presented for 2⩽ L y ⩽4. In the limit of infinite length, we calculate the ground state degeneracy per site (exponent of the ground state entropy), W( q). Generalizing q from Z + to C , we determine the continuous locus B in the complex q plane where W( q) is singular. We find the interesting result that for all L y values considered, the maximal point at which B crosses the real q-axis, denoted q c , is the same, and is equal to the value for the infinite square lattice, q c =3. This is the first family of strip graphs of which we are aware that exhibits this type of universality of q c .

Structural properties of Potts model partition functions and chromatic polynomials for lattice strips

The q-state Potts model partition function (equivalent to the Tutte polynomial) for a lattice strip of fixed width L y and arbitrary length L x has the form Z(G,q,v)=∑ j=1 N Z,G,λ c Z,G,j(λ Z,G,j) L x , where v is a temperature-dependent variable. The special case of the zero-temperature antiferromagnet ( v=−1) is the chromatic polynomial P( G, q). Using coloring and transfer matrix methods, we give general formulas for C X,G=∑ j=1 N X,G,λ c X,G,j for X= Z, P on cyclic and Möbius strip graphs of the square and triangular lattice. Combining these with a general expression for the (unique) coefficient c Z, G, j of degree d in q: c (d)=U 2d( q /2) , where U n ( x) is the Chebyshev polynomial of the second kind, we determine the number of λ Z, G, j 's with coefficient c ( d) in Z( G, q, v) for these cyclic strips of width L y to be n Z(L y,d)=(2d+1)(L y+d+1) −1 2L y L y−d for 0⩽ d⩽ L y and zero otherwise. For both cyclic and Möbius strips of these lattices, the total number of distinct eigenvalues λ Z, G, j is calculated to be N Z,L y,λ = 2L y L y . Results are also presented for the analogous numbers n P ( L y , d) and N P, L y , λ for P( G, q). We find that n P ( L y ,0)= n P ( L y −1,1)= M L y −1 (Motzkin number), n Z ( L y ,0)= C L y (the Catalan number), and give an exact expression for N P, L y , λ . Our results for N Z, L y , λ and N P, L y , λ apply for both the cyclic and Möbius strips of both the square and triangular lattices; we also point out the interesting relations N Z, L y , λ =2 N DA, tri, L y and N P, L y , λ =2 N DA, sq, L y , where N DA, Λ, n denotes the number of directed lattice animals on the lattice Λ. We find the asymptotic growths N Z, L y , λ ∼ L y −1/24 L y and N P, L y , λ ∼ L y −1/23 L y as L y →∞. Some general geometric identities for Potts model partition functions are also presented.

Greedy lattice animals: negative values and unconstrained maxima

Let $\{X_v, v \in \mathbb{Z}^d\}$ be i.i.d. random variables, and $S(\xi) = \sum_{v \in \xi} X_v$ be the weight of a lattice animal $\xi$. Let $N_n = \max\{S(\xi) : |\xi| = n$ and $\xi$ contains the origin} and $G_n = \max\{S(\xi) : \xi \subseteq [-n,n]^d\}$. We show that, regardless of the negative tail of the distribution of $X_v$ , if $\mathbf{E}( X_v^+)^d (\log^+ X_v^+))^{d+a} 0$, then first, $\lim_n n^{-1} N_n = N$ exists, is finite and constant a.e.; and, second, there is a transition in the asymptotic behavior of $G_n$ depending on the sign of $N$: if $N > 0$ then $G_n \approx n^d$, and if $N 0$. The exact behavior of $G_n$ in this last case depends on the positive tail of the distribution of $X_v$; we show that if it is nontrivial and has exponential moments, then $G_n \approx \log n$, with a transition from $G_n \approx n^d$ occurring in general not as predicted by large deviations estimates. Finally, if $x^d(1 - F(x)) \to \infty \text{ as }x \to \infty$, then no transition takes place.

Cluster diversity and entropy on the percolation model: the lattice animal identification algorithm

We present an algorithm to identify and count different lattice animals (LA's) in the site-percolation model. This algorithm allows a definition of clusters based on the distinction of cluster shapes, in contrast with the well-known Hoshen-Kopelman algorithm, in which the clusters are differentiated by their sizes. It consists in coding each unit cell of a cluster according to the nearest neighbors (NN) and ordering the codes in a proper sequence. In this manner, a LA is represented by a specific code sequence. In addition, with some modification the algorithm is capable of differentiating between fixed and free LA's. The enhanced Hoshen-Kopelman algorithm [J. Hoshen, M. W. Berry, and K. S. Minser, Phys. Rev. E 56, 1455 (1997)] is used to compose the set of NN code sequences of each cluster. Using Monte Carlo simulations on planar square lattices up to 2000x2000, we apply this algorithm to the percolation model. We calculate the cluster diversity and cluster entropy of the system, which leads to the determination of probabilities associated with the maximum of these functions. We show that these critical probabilities are associated with the percolation transition and with the complexity of the system.

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, τ = 4.062 570(8), 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 τ>3.903 18. 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 ν = 0.641 15(5), for both animals and polygons enumerated by area. The mean perimeter of polygons of area n is also calculated. A number of new amplitude estimates are given.

Improved rosenbluth monte carlo scheme for cluster counting and lattice animal enumeration

We describe an algorithm for the Rosenbluth Monte Carlo enumeration of clusters and lattice animals. The method may also be used to calculate associated properties such as moments or perimeter multiplicities of the clusters. The scheme is an extension of the Rosenbluth method for growing polymer chains and is a simplification of a scheme reported earlier by one of the authors. The algorithm may be used to obtain a Monte Carlo estimate of the number of distinct lattice animals on any lattice topology. The method is validated against exact and Monte Carlo enumerations for clusters up to size 50, on a two dimensional square lattice and three dimensional simple cubic lattice. The method may be readily adapted to yield Boltzmann weighted averages over clusters.

The cluster structure in collapsing animals

In this paper I revisit the connection between edge percolation and the collapse transition in lattice animals. It was shown by Domb (1976 J. Phys. A: Math. Gen. 9 L141) that the critical percolation point is a θ-transition in a certain model of self-interacting animals. I extend this result by showing that the free energy of lattice animals in the cycle-contact model is non-analytic in contact- and cycle-activities at points other than the critical percolation point. This correspondence between percolation and collapsing animals suggests that the collapse transition in this model may be related to percolation. Contact-collapse in animals is then studied from a percolation of clusters of contacts perspective. I first argue that there is a critical activity for percolation of clusters of contacts, and then investigate this numerically.

Topological effects in ring polymers. II. Influence Of persistence length

The interplay of topological constraints and the persistence length of ring polymers in their own melt is investigated by means of dynamical Monte Carlo simulations of a three-dimensional lattice model. We ask if the results are consistent with an asymptotically regime where the rings behave like (compact) lattice animals in a self-consistent network of topological constraints imposed by neighboring rings. Tuning the persistence length provides an efficient route to increase the ring overlap required for this mean-field picture to hold: The effective Flory exponent for the ring size decreases down to nu less, similar1/3 with increasing persistence length. Evidence is provided for the emergence of one additional characteristic length scale d(t) approximately N0, only weakly dependent on the persistence length and much larger than the excluded volume screening length xi. At distances larger than d(t) the conformational properties of the rings are governed by the topological interactions; at smaller distances rings and their linear chain counterparts become similar. (At distances smaller than xi both architectures are identical.) However, the crossover between both limits is intricate and broad, as a detailed discussion of the local fractal dimension (e.g., obtained from the static structure factor) reveals. This is due to various crossover effects which we are unable to separate even for the largest ring size (N=1024) presented here. The increased topological interactions also influence the dynamical properties. Mean-square displacements and their distributions depend crucially on the ring overlap, and show evidence of the existence of additional size and time scales. The diffusion constant of the rings goes down from effectively D(N) approximately N-1.22 for flexible rings with low overlap to D(N) approximately N-1.68 for strongly overlapping semiflexible rings.