Related papers: On the exponential growth rates of lattice animals…
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 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…
We examine the percolation model in $\mathbb{Z}^d$ by an approach involving lattice animals, in which their relevant characteristic is surface-area-to-volume ratio. Two critical exponents are introduced. The first is related to the growth…
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 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 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…
The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a…
Let us consider subcritical Bernoulli percolation on a connected, transitive, infinite and locally finite graph. In this paper, we propose a new (and short) proof of the exponential decay property for the volume of clusters. We do not rely…
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 present an implementation of Redelemeier's algorithm for the enumeration of lattice animals in high dimensional lattices. The implementation is lean and fast enough to allow us to extend the existing tables of animal counts, perimeter…
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…
A spread-out lattice animal is a finite connected set of edges in $\{ \{x,y\} \subset \mathbb{Z}^d:0<||x-y||\le L \}$. A lattice tree is a lattice animal with no loops.The best estimate on the critical point $p_c$ so far was achieved by…
It is a central prediction of renormalisation group theory that the critical behaviours of many statistical mechanics models on Euclidean lattices depend only on the dimension and not on the specific choice of lattice. We investigate the…
We calculate exponential growth constants describing the asymptotic behavior of several quantities enumerating classes of orientations of arrow variables on the bonds of several types of directed lattice strip graphs $G$ of finite width and…
Percolation on a five-dimensional simple hypercubic (sc(5)) lattice with extended neighborhoods is investigated by means of extensive Monte Carlo simulations, using an effective single-cluster growth algorithm. The critical exponents,…
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 extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian…
We study the growth and isoperimetry of infinite clusters in slightly supercritical Bernoulli bond percolation on transitive nonamenable graphs under the $L^2$ boundedness condition ($p_c<p_{2\to 2}$). Surprisingly, we find that the volume…
We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice $\Lambda$ by $\ell$ bonds connecting the same adjacent vertices,…
Recent advances on the glass problem motivate reexamining classical models of percolation. Here, we consider the displacement of an ant in a labyrinth near the percolation threshold on cubic lattices both below and above the upper critical…