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

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 the origin to $x\in{\mathbb{Z}}^d$, the probability of a connection from the origin to $x$, and the generating functions for lattice trees or lattice animals containing the origin and $x$. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to $\mathit{const.}|x|^{2-d}$ as $|x|\to\infty$, for $d\geq 5$ for self-avoiding walk, for $d\geq19$ for percolation, and for sufficiently large $d$ for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349--408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if $d>4$) condition under which the two-point function of a random walk on ${{\mathbb{Z}}^d}$ is asymptotic to $\mathit{const.}|x|^{2-d}$ as $|x|\to\infty$.Comment: Published in at http://dx.doi.org/10.1214/009117907000000231 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Violating conformal invariance: Two-dimensional clusters grafted to wedges, cones, and branch points of Riemann surfaces

We present simulations of 2-d site animals on square and triangular lattices in non-trivial geomeLattice animals are one of the few critical models in statistical mechanics violating conformal invariance. We present here simulations of 2-d site animals on square and triangular lattices in non-trivial geometries. The simulations are done with the newly developed PERM algorithm which gives very precise estimates of the partition sum, yielding precise values for the entropic exponent $\theta$ ($Z_N \sim \mu^N N^{-\theta}$). In particular, we studied animals grafted to the tips of wedges with a wide range of angles $\alpha$, to the tips of cones (wedges with the sides glued together), and to branching points of Riemann surfaces. The latter can either have $k$ sheets and no boundary, generalizing in this way cones to angles $\alpha > 360$ degrees, or can have boundaries, generalizing wedges. We find conformal invariance behavior, $\theta \sim 1/\alpha$, only for small angles ($\alpha \ll 2\pi$), while $\theta \approx const -\alpha/2\pi$ for $\alpha \gg 2\pi$. These scalings hold both for wedges and cones. A heuristic (non-conformal) argument for the behavior at large $\alpha$ is given, and comparison is made with critical percolation.Comment: 4 pages, includes 3 figure

Series expansions of the percolation probability on the directed triangular lattice

We have derived long series expansions of the percolation probability for site, bond and site-bond percolation on the directed triangular lattice. For the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35. For the site-bond problem, which has not been studied before, we have derived the series to order 32. Our estimates of the critical exponent $\beta$ are in full agreement with results for similar problems on the square lattice, confirming expectations of universality. For the critical probability and exponent we find in the site case: $q_c = 0.4043528 \pm 0.0000010$ and $\beta = 0.27645 \pm 0.00010$; in the bond case: $q_c = 0.52198\pm 0.00001$ and $\beta = 0.2769\pm 0.0010$; and in the site-bond case: $q_c = 0.264173 \pm 0.000003$ and $\beta = 0.2766 \pm 0.0003$. In addition we have obtained accurate estimates for the critical amplitudes. In all cases we find that the leading correction to scaling term is analytic, i.e., the confluent exponent $\Delta = 1$.Comment: 26 pages, LaTeX. To appear in J. Phys.

On directed interacting animals and directed percolation

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 animals) as special cases and its collapse transition is equivalent to a directed bond percolation threshold. Precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n=17. The crossover region in the vicinity of the collapse transition is analyzed in detail and the crossover exponent $\phi$ is determined directly from the singular part of the free energy. We show using scaling arguments and an exact relation due to Dhar that $\phi$ is equal to the Fisher exponent $\sigma$ governing the size distribution of large directed percolation clusters.Comment: 23 pages, 3 figures; J. Phys. A 35 (2002) 272

Limiting shape for directed percolation models

We consider directed first-passage and last-passage percolation on the nonnegative lattice Z_+^d, d\geq2, with i.i.d. weights at the vertices. Under certain moment conditions on the common distribution of the weights, the limits g(x)=lim_{n\to\infty}n^{-1}T(\lfloor nx\rfloor) exist and are constant a.s. for x\in R_+^d, where T(z) is the passage time from the origin to the vertex z\in Z_+^d. We show that this shape function g is continuous on R_+^d, in particular at the boundaries. In two dimensions, we give more precise asymptotics for the behavior of g near the boundaries; these asymptotics depend on the common weight distribution only through its mean and variance. In addition we discuss growth models which are naturally associated to the percolation processes, giving a shape theorem and illustrating various possible types of behavior with output from simulations.Comment: Published at http://dx.doi.org/10.1214/009117904000000838 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Simulations of lattice animals and trees

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 according to the animal (tree) ensemble, and prune or branch the further growth according to a heuristic fitness function. In contrast to previous applications of PERM, this fitness function is {\it not} the weight with which the actual configuration would contribute to the partition sum, but is closely related to it. We obtain high statistics of animals with up to several thousand sites in all dimension 2 <= d <= 9. In addition to the partition sum (number of different animals) we estimate gyration radii and numbers of perimeter sites. In all dimensions we verify the Parisi-Sourlas prediction, and we verify all exactly known critical exponents in dimensions 2, 3, 4, and >= 8. In addition, we present the hitherto most precise estimates for growth constants in d >= 3. For clusters with one site attached to an attractive surface, we verify the superuniversality of the cross-over exponent at the adsorption transition predicted by Janssen and Lyssy. Finally, we discuss the collapse of animals and trees, arguing that our present version of the algorithm is also efficient for some of the models studied in this context, but showing that it is {\it not} very efficient for the `classical' model for collapsing animals.Comment: 17 pages RevTeX, 29 figures include