Lattice Animals Research Articles

Percolation in random sequential adsorption of lattice animals on a three-dimensional cubic lattice

Percolation properties of the Random Sequential Adsorption (RSA) of objects of various shapes on simple three-dimensional (3D) cubic lattice are studied numerically by means of Monte Carlo simulations. Depositing objects are “lattice animals”, made of a certain number of nearest neighbor sites on a lattice. The aim of this work is to investigate the impact of the geometrical properties of the shapes on the values of percolation threshold θp∗. We analyzed all lattice animals of size n⩽5.Thanks to an extensive database of studied objects, we found that the number of nearest neighbors N1 and the radius of gyration Rg of the objects are correlated with the values of percolation threshold θp∗. For lattice animals of the same size, the percolation threshold θp∗ decreases with an increase in the number of the object’s nearest neighbors N1. If objects of the same size n have the same number of nearest neighbors N1, their percolation threshold θp∗ decreases with an increase in the radius of gyration Rg.

2D lattice animal formation in the irreversible aggregation of patchy particles

With the self-assembly of patchy colloids, we can engineer many complex structures. In the present work, we are modeling spherical particles with 2 diagonally opposite patches using the Kern-Frenkel potential coupled with isotropic square well potential having different interaction ranges. The isotropic square well potential formed reversible bonds while the patches were irreversibly bonded. A state diagram is calculated where we observed a range of structures from a 1D chain, 2D planar cluster, and 3D random clusters depending on the width of the patch. These structures are controlled by the number of bonds each patch can form for a given patch angle. The range of the patch width can be analytically calculated using geometric arguments to identify different types of clusters, which agree with our simulations. The aggregation process was diffusion-limited, forming clusters with fractal dimension 2 in the low volume fraction limit, otherwise known as flocculation region, and 2.5 in the high volume fraction limit or percolation region, irrespective of the type of clusters formed. In the flocculation limit of the DLCA aggregation model, we expect a fractal dimension of 1.8, contrary to that, we observed the formation of lattice animals having a fractal dimension of 2. The presence of lattice animals is confirmed using the scaling laws of size distribution and chemical length of the clusters formed in the flocculation limit.

Spread-out limit of the critical points for lattice trees and lattice animals in dimensions

AbstractA spread-out lattice animal is a finite connected set of edges in $\{\{x,y\}\subset \mathbb{Z}^d\;:\;0\lt \|x-y\|\le L\}$. A lattice tree is a lattice animal with no loops. The best estimate on the critical point $p_{\textrm{c}}$ so far was achieved by Penrose (J. Stat. Phys. 77, 3–15, 1994) : $p_{\textrm{c}}=1/e+O(L^{-2d/7}\log L)$ for both models for all $d\ge 1$. In this paper, we show that $p_{\textrm{c}}=1/e+CL^{-d}+O(L^{-d-1})$ for all $d\gt 8$, where the model-dependent constant $C$ has the random-walk representation \begin{align*} C_{\textrm{LT}}=\sum _{n=2}^\infty \frac{n+1}{2e}U^{*n}(o),&& C_{\textrm{LA}}=C_{\textrm{LT}}-\frac 1{2e^2}\sum _{n=3}^\infty U^{*n}(o), \end{align*}where $U^{*n}$ is the $n$-fold convolution of the uniform distribution on the $d$-dimensional ball $\{x\in{\mathbb R}^d\;: \|x\|\le 1\}$. The proof is based on a novel use of the lace expansion for the 2-point function and detailed analysis of the 1-point function at a certain value of $p$ that is designed to make the analysis extremely simple.

On the exponential growth rates of lattice animals and interfaces

AbstractWe introduce a formula for translating any upper bound on the percolation threshold of a lattice$G$into a lower bound on the exponential growth rate of lattice animals$a(G)$and vice versa. We exploit this in both directions. We obtain the rigorous lower bound${\dot{p}_c}({\mathbb{Z}}^3)\gt 0.2522$for 3-dimensional site percolation. We also improve on the best known asymptotic bounds on$a({\mathbb{Z}}^d)$as$d\to \infty$. Our formula remains valid if instead of lattice animals we enumerate certain subspecies called interfaces. Enumerating interfaces leads to functional duality formulas that are tightly connected to percolation and are not valid for lattice animals, as well as to strict inequalities for the percolation threshold.Incidentally, we prove that the rate of the exponential decay of the cluster size distribution of Bernoulli percolation is a continuous function of$p\in (0,1)$.

Correlation length of the two-dimensional random field Ising model via greedy lattice animal

For the two-dimensional random field Ising model where the random field is given by independent and identically distributed mean zero Gaussian variables with variance ε2, we study (one natural notion of) the correlation length, which is the critical size of a box at which the influences of the random field and of the boundary condition on the spin magnetization are comparable. We show that as ε→0, at zero temperature the correlation length scales as eΘ(ε−4∕3) (and our upper bound applies for all positive temperatures).

Enumerating Stable Nanopores in Graphene and Their Geometrical Properties Using the Combinatorics of Hexagonal Lattices

Nanopores in two-dimensional (2D) materials, including graphene, can be used for a variety of applications, such as gas separations, water desalination, and DNA sequencing. So far, however, all plausible isomeric shapes of graphene nanopores have not been enumerated. Instead, a probabilistic approach has been followed to predict nanopore shapes in 2D materials, due to the exponential increase in the number of nanopores as the size of the vacancy increases. For example, there are 12 possible isomers when N = 6 atoms are removed, a number that theoretically increases to 11.7 million when N = 20 atoms are removed from the graphene lattice. In this regard, the development of a smaller, exhaustive data set of stable nanopore shapes can help future experimental and theoretical studies focused on using nanoporous 2D materials in various applications. In this work, we use the theory of 2D triangular "lattice animals" to create a library of all stable graphene nanopore shapes based on a modification of a well-known algorithm in the mathematical combinatorics of polyforms known as Redelmeier's algorithm. We show that there exists a correspondence between graphene nanopores and triangular polyforms (called polyiamonds) as well as hexagonal polyforms (called polyhexes). We develop the concept of a polyiamond ID to identify unique nanopore isomers. We also use concepts from polyiamond and polyhex geometries to eliminate unstable nanopores containing dangling atoms, bonds, and moieties. We verify using density functional theory calculations that such pores are indeed unstable. The exclusion of these unstable nanopores leads to a remarkable reduction in the possible nanopores from 11.7 million for N = 20 to only 0.184 million nanopores, thereby indicating that the number of stable nanopores is almost 2 orders of magnitude lower and is much more tractable. Not only that, by extracting the polyhex outline, our algorithm allows searching for nanopores with dimensions and shape factors in a specified range, thus aiding the design of the geometrical properties of nanopores for specific applications. We also provide the coordinate files of the stable nanopores as a library to facilitate future theoretical studies of these nanopores.

Adsorption of Lattice Polymers with Quenched Topologies

We introduce a framework for adsorption of a single polymer in which the topology of the polymer is quenched before adsorption, in contrast to more standard adsorption models having annealed topology. Our "topology" refers either to the precise branching structure of a branched polymer (in any dimension), or else to the knot type of a ring polymer in three dimensions. The quenched topology is chosen uniformly at random from all lattice polymers of a given size in one of four classes (lattice animals, trees, combs, or rings), and we then consider adsorption of the subclass of configurations that have the quenched topology. When the polymer-surface attraction increases without bound, the quenched topological structure keeps a macroscopic fraction of monomers off the surface, in contrast with annealed models that asymptotically have 100% of monomers in the surface. We prove properties of the limiting free energy and the critical point in each model, although important open questions remain. We pay special attention to the class of comb polymers, which admit some rigorous answers to questions that otherwise remain open. Since the class of all combs was not previously examined rigorously in full generality, we also prove the existence of its growth constant and its limiting free energy for annealed adsorption.

Minimum-Perimeter Lattice Animals and the Constant-Isomer Conjecture

We consider minimum-perimeter lattice animals, providing a set of conditions which are sufficient for a lattice to have the property that inflating all minimum-perimeter animals of a certain size yields (without repetitions) all minimum-perimeter animals of a new, larger size. We demonstrate this result on the two-dimensional square and hexagonal lattices. In addition, we characterize the sizes of minimum-perimeter animals on these lattices that are not created by inflating members of another set of minimum-perimeter animals.

Algorithms for Counting Minimum-Perimeter Lattice Animals

Algorithms for Counting Minimum-Perimeter Lattice Animals

Automatic generation of formulae for polyominoes with a fixed perimeter defect

A polyomino is a shape best described as a connected set of cells in the square lattice. As part of recreational mathematics, polyominoes have seen active research since the 1950s. Simultaneously, polyominoes have been investigated in statistical physics under the name “lattice animals,” mainly in regards to percolation problems. One of the main points of interest is to solve the yet unanswered question of how many different polyominoes exist. Most of the focus, so far, went to estimating the number of different polyominoes that can be made with a given fixed number of cells. Recently, there are increased efforts to discover the number of polyominoes with not only a given area, but with a given perimeter size or perimeter defect as well.Roughly speaking, the perimeter defect is a number that measures how many twists a polyomino has. Formally, the defect of a polyomino P is defined as the deviation of the perimeter size of P from the maximum possible perimeter size taken over all polyominoes of the same area as P. Interestingly, a polyomino might contain a column or row which can be “cut” out of the polyomino, then the remaining parts of the polyomino are “glued” back together along the cut, and result in a smaller polyomino with equal perimeter defect to the original. By repeating such “cut-and-glue” operations on a polyomino until all matching columns and rows have been removed, one obtains a so-called “reduced polyomino.”We expand on the efforts regarding perimeter defect and reduced polyominoes, in two directions. First, given a fixed perimeter defect, we demonstrate and prove upper and lower bounds on the width and height, as well as an upper bound on the area, of any reduced polyomino with a given perimeter defect. Second, we present an algorithm for enumerating all reduced polyominoes with a given perimeter defect, and calculate their combined generating function. From the generating function, we can extract a formula A(k)(n) for the number of polyominoes that have the fixed perimeter defect k and area n, for any n. Using our new algorithm, and in the case of k=5 some additional manual calculations, we provide closed formulae of A(k)(n), for up to k=5, as well as the generating functions for up to k=5. This is an improvement over the previously known formulae, which were known only up to k=3.

Simulation study of random sequential deposition of binary mixtures of lattice animals on a three-dimensional cubic lattice

Random sequential adsorption of mixtures of objects of various shapes on a three-dimensional (3D) cubic lattice is studied numerically by means of Monte Carlo simulations. Depositing objects are ‘lattice animals’, made of a certain number of nearest neighbor sites on a lattice. We analyzed binary mixtures composed of shapes of equal size, n = 3, 4, 5. We concentrate here on the influence of geometrical properties of the shapes on the jamming coverage θ J and on the temporal evolution of the density θ(t). The approach of the coverage θ(t) to the jamming limit θ J is found to be exponential, θ J − θ(t) ∼ exp(−t/σ), both for the mixtures and their components. The values of the relaxation time σ are determined by the number of different orientations m that lattice animals can take when placed on a cubic lattice. The value of the relaxation time σ for a mixture is approximately twice the relaxation time for the pure component shape with a larger number m of possible orientations. Depending on the local geometry of the objects making the mixture, the jamming coverage of a mixture θ J can be either greater than both single-component jamming coverages or it can be in between these values. The first case is the most common, while in the second case, the jamming density of the mixture is very close to the higher jamming density for the pure component shapes. For a majority of the investigated mixtures, a component with a larger number of orientations m has a larger value of the fractional jamming density.

Correction to: NoBLE for Lattice Trees and Lattice Animals

Correction to: NoBLE for Lattice Trees and Lattice Animals

NoBLE for Lattice Trees and Lattice Animals

We study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice {mathbb {Z}}^d in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension 16 and 17, respectively. Such results have previously been obtained by Hara and Slade in sufficiently high dimensions. The dimension above which their results apply was not yet specified. We rely on the non-backtracking lace expansion (NoBLE) method that we have recently developed. The NoBLE makes use of an alternative lace expansion for LAs and LTs that perturbs around non-backtracking random walk rather than around simple random walk, leading to smaller corrections. The NoBLE method then provides a careful computational analysis that improves the dimension above which the result applies. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to d_c=8 for both models, as is known for sufficiently spread-out models by the results of Hara and Slade mentioned earlier. The main ingredients in this paper are (a) a derivation of a non-backtracking lace expansion for the LT and LA two-point functions; (b) bounds on the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) sharp numerical bounds on the coefficients. Our proof is complemented by a computer-assisted numerical analysis that verifies that the necessary bounds used in the NoBLE are satisfied.

Transverse size of interacting directed lattice animals studied by Yang–Lee approach

Distributions of zeros of the grand canonical partition function in the complex fugacity plane are examined numerically for a model of self interacting directed lattice animals on infinite strips of the square lattice. It is shown that patterns of these zeros have a simple circular-like forms for large values of the interaction strength, with a small fraction of points scattered over the plane. As the interaction weakens, the zeros make up a more complicated structure which consists of a large number of branches. For weak interaction strength, the zeros lying closest to the real axis approach it by following the power-law behavior as the function of strip width. This allowed us to get quite good estimate of transverse size critical exponent for animals in the swollen state as well as at the point of their collapse transition. For stronger interactions, we found that these zeros move much faster toward the real axis—following an exponential law, which leads us to conjecture that animal transverse size diverges in a logarithmic way in the thermodynamic limit. An additional analysis, based on finite-size scaling analysis of the animal longitudinal correlation length, supports this conjecture.

Percolation thresholds on high-dimensional D_{n} and E_{8}-related lattices.

The site and bond percolation problems are conventionally studied on (hyper)cubic lattices, which afford straightforward numerical treatments. The recent implementation of efficient simulation algorithms for high-dimensional systems now also facilitates the study of D_{n} root lattices in n dimensions as well as E_{8}-related lattices. Here, we consider the percolation problem on D_{n} for n=3 to 13 and on E_{8} relatives for n=6 to 9. Precise estimates for both site and bond percolation thresholds obtained from invasion percolation simulations are compared with dimensional series expansion based on lattice animal enumeration for D_{n} lattices. As expected, the bond percolation threshold rapidly approaches the Bethe lattice limit as n increases for these high-connectivity lattices. Corrections, however, exhibit clear yet unexplained trends. Interestingly, the finite-size scaling exponent for invasion percolation is found to be lattice and percolation-type specific.

Sublinear variance in Euclidean first-passage percolation

The Euclidean first-passage percolation model of Howard and Newman is a rotationally invariant percolation model built on a Poisson point process. It is known that the passage time between 0 and ne1 obeys a diffusive upper bound: VarT(0,ne1)≤Cn, and in this paper we improve this inequality to Cn∕logn. The methods follow the strategy used for sublinear variance proofs on the lattice, using the Falik–Samorodnitsky inequality and a Bernoulli encoding, but with substantial technical difficulties. To deal with the different setup of the Euclidean model, we represent the passage time as a function of Bernoulli sequences and uniform sequences, and develop several “greedy lattice animal” arguments.

Random sequential adsorption of lattice animals on a three-dimensional cubic lattice

The properties of the random sequential adsorption of objects of various shapes on simple three-dimensional (3D) cubic lattice are studied numerically by means of Monte Carlo simulations. Depositing objects are "lattice animals," made of a certain number of nearest-neighbor sites on a lattice. The aim of this work is to investigate the impact of the geometrical properties of the shapes on the jamming density θ_{J} and on the temporal evolution of the coverage fraction θ(t). We analyzed all lattice animals of size n=1, 2, 3, 4, and 5. A significant number of objects of size n⩾6 were also used to confirm our findings. Approach of the coverage θ(t) to the jamming limit θ_{J} is found to be exponential, θ_{J}-θ(t)∼exp(-t/σ), for all lattice animals. It was shown that the relaxation time σ increases with the number of different orientations m that lattice animals can take when placed on a cubic lattice. Orientations of the lattice animal deposited in two randomly chosen places on the lattice are different if one of them cannot be translated into the other. Our simulations performed for large collections of 3D objects confirmed that σ≅m∈{1,3,4,6,8,12,24}. The presented results suggest that there is no correlation between the number of possible orientations m of the object and the corresponding values of the jamming density θ_{J}. It was found that for sufficiently large objects, changing of the shape has considerably more influence on the jamming density than increasing of the object size.

Equilibrium size distribution and phase separation of multivalent, molecular assemblies in dilute solution.

Multivalent molecules can bind a limited number of multiple neighbors via specific interactions. In this paper, we investigate theoretically the self-assembly and phase separation of such molecules in dilute solution. We show that the equilibrium size (n) distributions of linear or branched assemblies qualitatively differ; the former decays exponentially with the relative size n/N[combining macron] (N[combining macron] = n), while the latter decays as a power law, with an exponential cutoff only for n ⪆ N[combining macron]2 ≫ N[combining macron]. In some cases, finite, branched assemblies are unstable and show a sol-gel transition at a critical concentration. In dilute solutions, non-specific interactions result in phase separation, whose critical point is described by an effective Flory Huggins theory that is sensitive to the nature of these distributions.

Percolation games, probabilistic cellular automata, and the hard-core model

Let each site of the square lattice $$\mathbb {Z}^2$$ be independently assigned one of three states: a trap with probability p, a target with probability q, and open with probability $$1-p-q$$ , where $$0<p+q<1$$ . Consider the following game: a token starts at the origin, and two players take turns to move, where a move consists of moving the token from its current site x to either $$x+(0,1)$$ or $$x+(1,0)$$ . A player who moves the token to a trap loses the game immediately, while a player who moves the token to a target wins the game immediately. Is there positive probability that the game is drawn with best play—i.e. that neither player can force a win? This is equivalent to the question of ergodicity of a certain family of elementary one-dimensional probabilistic cellular automata (PCA). These automata have been studied in the contexts of enumeration of directed lattice animals, the golden-mean subshift, and the hard-core model, and their ergodicity has been noted as an open problem by several authors. We prove that these PCA are ergodic, and correspondingly that the game on $$\mathbb {Z}^2$$ has no draws. On the other hand, we prove that certain analogous games do exhibit draws for suitable parameter values on various directed graphs in higher dimensions, including an oriented version of the even sublattice of $$\mathbb {Z}^d$$ in all $$d\ge 3$$ . This is proved via a dimension reduction to a hard-core lattice gas in dimension $$d-1$$ . We show that draws occur whenever the corresponding hard-core model has multiple Gibbs distributions. We conjecture that draws occur also on the standard oriented lattice $$\mathbb {Z}^d$$ for $$d\ge 3$$ , but here our method encounters a fundamental obstacle.

Series expansion of the percolation threshold on hypercubic lattices

We study proper lattice animals for bond- and site-percolation on the hypercubic lattice to derive asymptotic series of the percolation threshold pc in 1/d, The first few terms of these series were computed in the 1970s, but the series have not been extended since then. We add two more terms to the series for and one more term to the series for , using a combination of brute-force enumeration, combinatorial identities and an approach based on Padé approximants, which requires much fewer resources than the classical method. We discuss why it took 40 yr to compute these terms, and what it would take to compute the next ones. En passant, we present new perimeter polynomials for site and bond percolation and numerical values for the growth rate of bond animals.