Related papers: Counting Lattice Animals in High Dimensions
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, $\tau = 4.062570(8)$, for the…
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 new scheme…
We perform numerical simulations of the lattice-animal problem at the upper critical dimension d=8 on hypercubic lattices in order to investigate logarithmic corrections to scaling there. Our stochastic sampling method is based on the…
We have developed an improved algorithm that allows us to enumerate the number of site animals on the square lattice up to size 46. We also calculate the number of lattice trees up to size 44 and the radius of gyration of both lattice…
We describe Maple packages for the automatic generation of generating functions(and series expansions) for counting lattice animals(fixed polyominoes), in the two-dimensional hexagonal lattice, of bounded but arbitrary width. Our Maple…
We consider minimal-perimeter lattice animals, providing a set of conditions which are sufficient for a lattice to have the property that inflating all minimal-perimeter animals of a certain size yields (without repetitions) all…
We compute the first three terms of the 1/d expansions for the growth constants and one-point functions of nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd, with rigorous error estimates. The proof uses…
A straightforward algorithm for the symbolic computation of higher-order symmetries of nonlinear evolution equations and lattice equations is presented. The scaling properties of the evolution or lattice equations are used to determine the…
We introduce a method for translating any upper bound on the percolation threshold of a lattice $G$ into a lower bound on the exponential growth rate $a(G)$ of lattice animals and vice-versa. We exploit this in both directions. We improve…
We study proper lattice animals for bond- and site-percolation on the hypercubic lattice $\mathbb{Z}^d$ to derive asymptotic series of the percolation threshold $p_c$ in $1/d$, The first few terms of these series were computed in the 1970s,…
We study the statistics of column-convex lattice animals resulting from the stacking of squares on a single or double staircase. We obtain exact expressions for the number of animals with a given length and area, their mean length and their…
We 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 to improve on the best known…
Density estimates based on point processes are often restrained to regions with irregular boundaries or holes. We propose a density estimator, the lattice-based density estimator, which produces reasonable density estimates under these…
We propose and test an algorithm to simulate a lattice system of interacting fermions in two spatial dimensions. The approach is an extension of the entanglement renormalization technique [Phys. Rev. Lett. 99, 220405 (2007)] and the related…
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…
We present a new and more efficient implementation of transfer-matrix methods for exact enumerations of lattice objects. The new method is illustrated by an application to the enumeration of self-avoiding polygons on the square lattice. A…
In order to investigate the dependence on lattice size of several observables in percolation, the Hoshen-Kopelman algorithm was modified so that growing lattices could be simulated. By this way, when simulating a lattice of size L, lattices…
A novel approach to parallelize the well-known Hoshen-Kopelman algorithm has been chosen, suitable for simulating huge lattices in high dimensions on massively-parallel computers with distributed memory and message passing. This method…
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…
We introduce new combinatorial (bijective) methods that enable us to compute the average value of three parameters of directed animals of a given area, including the site perimeter. Our results cover directed animals of any one-line source…