Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals

How long does a self-avoiding walk on a discrete $d$-dimensional torus have to be before it begins to behave differently from a self-avoiding walk on $\mathbb{Z}^d$? We consider a version of this question for weakly self-avoiding walk on a torus in dimensions $d>4$. On $\mathbb{Z}^d$ for $d>4$, the partition function for $n$-step weakly self-avoiding walk is known to be asymptotically purely exponential, of the form $A\mu^n$, where $\mu$ is the growth constant for weakly self...

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 rate in size of the number of lattice animals up to translation whose surface-area-to-volume ratio is marginally greater than $1/p_c -1$. The second describes how unusually large clusters form in the percolation model at parameter values sli...

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...

We define a random graph obtained via connecting each point of $\mathbb{Z}^d$ independently to a fixed number $1 \leq k \leq 2d$ of its nearest neighbors via a directed edge. We call this graph the directed $k$-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional $k$-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed $k$-neighbor graph between them in at least one, ...

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 on the best known asymptotic lower and upper bounds on $a(\mathbb{Z}^d)$ as $d\to \infty$. We use percolation as a tool to obtain the latter, and conversely we use the former to obtain lower bounds on $p_c(\mathbb{Z}^d)$. We obtain the rigor...

We consider random walks (RWs) and self-avoiding walks (SAWs) on disordered lattices directly at the percolation threshold. Applying numerical simulations, we study the scaling behavior of the models on the incipient percolation cluster in space dimensions d=2, 3, 4. Our analysis yields estimates of universal exponents, governing the scaling laws for configurational properties of RWs and SAWs.

In long-range percolation on $\mathbb{Z}^d$, we connect each pair of distinct points $x$ and $y$ by an edge independently at random with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $\alpha>0$ is fixed and $\beta\geq 0$ is a parameter. In a previous paper, we proved that if $0<\alpha<d$ then the critical two-point function satisfies the spatially averaged upper bound \[ \frac{1}{r^d}\sum_{x\in [-r,r]^d} \mathbb{P}_{\beta_c}(0\leftrightarrow x) \preceq r^{-d+\alpha} ...

Let Z_N be the number of self-avoiding paths of length N starting from the origin on the infinite cluster obtained after performing Bernoulli percolation on Z^d with parameter p>p_c(Z^d). The object of this paper is to study the connective constant of the dilute lattice \limsup_{N\to \infty} Z_N^{1/N}, which is a non-random quantity. We want to investigate if the inequality \limsup_{N\to \infty} (Z_N)^{1/N} \le \lim_{N\to \infty} E[Z_N]^{1/N} obtained with the Borel-Cantelli ...

We relate $\phi(\bf{x},s)$, the average number of sites at a transverse distance $\bf{x}$ in the directed animals with $s$ sites in $d$ transverse dimensions, to the two-point correlation function of a lattice gas with nearest neighbor exclusion in $d$ dimensions. For large $s$, $\phi(\bf{x},s)$ has the scaling form $\frac{s}{R_s^d} f(|\bf{x}|/R_s)$, where $R_s$ is the root mean square radius of gyration of animals of $s$ sites. We determine the exact scaling function for $d ...

This is the first of two papers on the critical behaviour of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents eta and delta, for the nearest-neighbour model in very high dimensions d>>6 and for sufficiently spread-out models in all dimensions d>6. The exponent eta describes the low frequency behaviour of the Fourier transform of the critical two-point connectivity function, while delta describes the behaviour ...