Related papers: Haruspicy 3: The directed bond-animal generating f…
This article introduces a new, simple solvable lattice for directed animals: the directed king's lattice, or square lattice with next nearest neighbor bonds and preferred directions {W, NW, N, NE, E}. We show that the directed animals in…
A directed animal is a percolation cluster in the directed site percolation model. The aim of this paper is to exhibit a strong relation between the problem of computing the generating function $\G$ of directed animals on the square…
We prove that the anisotropic generating function of self-avoiding polygons is not a D-finite function - proving a conjecture of Guttmann and Enting. This result is also generalised to self-avoiding polygons on hypercubic lattices. Using…
We consider the local limit of finite uniformly distributed directed animals on the square lattice viewed from the root. Two constructions of the resulting uniform infinite directed animal are given: one as a heap of dominoes, constructed…
We continue the investigations of lattice walks in the three dimensional lattice restricted to the positive octant. We separate models which clearly have a D-finite generating function from models for which there is no reason to expect that…
In this paper, we revisit the enumeration of directed animals using gas models. We show that there exists a natural construction of random directed animals on any directed graph together with a particle system that explains at the level of…
We study directed rigidity percolation (equivalent to directed bootstrap percolation) on three different lattices: square, triangular, and augmented triangular. The first two of these display a first-order transition at p=1, while the…
We prove that the growth constants for nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd are asymptotic to 2de as the dimension goes to infinity, and that their critical one-point functions converge to e.…
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…
Stacked directed animals and multi-directed animals are two lattice models defined by Bousquet-M\'{e}lou and Rechnitzer in 2002. The original definitions of those models involve heaps of pieces, i.e., some geometric representation of…
We investigate site and bond percolation in triangular and square lattices subjected to linear distortion. In contrast to previously studied distortion schemes that preserve lattice geometry, linear distortion dislocates regular lattice…
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 show that the problem of directed percolation on an arbitrary lattice is equivalent to the problem of m directed random walkers with rather general attractive interactions, when suitably continued to m=0. In 1+1 dimensions, this is dual…
A new algorithm for the derivation of low-density series for percolation on directed lattices is introduced and applied to the square lattice bond and site problems. Numerical evidence shows that the computational complexity grows…
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…
We present high statistics simulations of weighted lattice bond animals and lattice trees on the square lattice, with fugacities for each non-bonded contact and for each bond between two neighbouring monomers. The simulations are performed…
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 consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ${\mathbb{Z}}^d$. The two-point functions of these models are respectively the generating function for self-avoiding walks from…
We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on the d-dimensional hyper cubic lattice having long finite-range connections, above their upper critical dimensions d=4…
We study mixed site-bond directed percolation on 2D and 3D lattices by using time-dependent simulations. Our results are compared with rigorous bounds recently obtained by Liggett and by Katori and Tsukahara. The critical fractions…