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

We consider the spread-out models of the self-avoiding walk and its finite-memory version, called the memory-$\tau$ walk. For both models, each step is uniformly distributed over the d-dimensional box $\{x\in\mathbb Z^d:\|x\|_{\infty} \le L\}$. The critical point $p_c^\tau$ for the memory-$\tau$ walk is increasing in $\tau$ and converges to the critical point $p_c^\infty$ for the self-avoiding walk as $\tau\uparrow\infty$. The best estimate of the rate of convergence so far w...

We prove $|x|^{-2}$ decay of the critical two-point function for the continuous-time weakly self-avoiding walk on $\mathbb{Z}^d$, in the upper critical dimension $d=4$. This is a statement that the critical exponent $\eta$ exists and is equal to zero. Results of this nature have been proved previously for dimensions $d \geq 5$ using the lace expansion, but the lace expansion does not apply when $d=4$. The proof is based on a rigorous renormalisation group analysis of an exact...

We consider percolation on $\mathbb{Z}^d$ and on the $d$-dimensional discrete torus, in dimensions $d \ge 11$ for the nearest-neighbour model and in dimensions $d>6$ for spread-out models. For $\mathbb{Z}^d$, we employ a wide range of techniques and previous results to prove that there exist positive constants $c$ and $C$ such that the slightly subcritical two-point function and one-arm probabilities satisfy \[ \mathbb{P}_{p_c-\varepsilon}(0 \leftrightarrow x) \leq \frac{C}{\...

In this paper we investigate the survival probability, \theta_n, in high-dimensional statistical physical models, where \theta_n denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if a certain self-repellence condition is satisfied, then n\theta_n\ra 2/(AV), where A is the asymptotic expected number of particles alive at time n, and V is the vertex factor of...

We use the lace expansion to give a simple proof that the critical two-point function for weakly self-avoiding walk on $\mathbb{Z}^d$ has decay $|x|^{-(d-2)}$ in dimensions $d>4$. The proof uses elementary Fourier analysis and the Riemann--Lebesgue Lemma.

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, but the series have not been extended since then. We add two more terms to the series for $\pcsite$ and one more term to the series for $\pcbond$, using a combination of brute-force enumeration, combinatorial identities and an approach bas...

In this paper, we study the abundance of self-avoiding paths of a given length on a supercritical percolation cluster on $\bbZ^d$. More precisely, we count $Z_N$ the number of self-avoiding paths of length $N$ on the infinite cluster, starting from the origin (that we condition to be in the cluster). We are interested in estimating the upper growth rate of $Z_N$, $\limsup_{N\to \infty} Z_N^{1/N}$, that we call the connective constant of the dilute lattice. After proving that ...

We consider random interlacements on Z^d, with d bigger or equal to 3, when their vacant set is in a strongly percolative regime. We derive an asymptotic upper bound on the probability that the random interlacements disconnect a box of large side-length from the boundary of a larger homothetic box. As a corollary, we obtain an asymptotic upper bound on a similar quantity, where the random interlacements are replaced by the simple random walk. It is plausible, but open at the ...

We prove two results on the delocalization of the endpoint of a uniform self-avoiding walk on Z^d for d>1. We show that the probability that a walk of length n ends at a point x tends to 0 as n tends to infinity, uniformly in x. Also, for any fixed x in Z^d, this probability decreases faster than n^{-1/4 + epsilon} for any epsilon >0. When |x|= 1, we thus obtain a bound on the probability that self-avoiding walk is a polygon.

We study independent long-range percolation on $\mathbb{Z}^d$ for all dimensions $d$, where the vertices $u$ and $v$ are connected with probability 1 for $\|u-v\|_\infty=1$ and with probability $p(\beta,\{u,v\})=1-e^{-\beta \int_{u+\left[0,1\right)^d} \int_{v+\left[0,1\right)^d} \frac{1}{\|x-y\|_2^{2d}}d x d y } \approx \frac{\beta}{\|u-v\|_2^{2d}}$ for $\|u-v\|_\infty \geq 2$. Let $u \in \mathbb{Z}^d$ be a point with $\|u\|_\infty=n$. We show that both the graph distance $D(...