Infinite canonical super-Brownian motion and scaling limits

We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on $\Z^d$ when these objects are (a) critical; (b) mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We conjecture that it also arises as the scaling limit in various other models above the upper-critical dimension, such as the incipient infinite lattice tree above 8 dimensions, the incipient infinite cluster for unoriented percolation, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions. This paper also serves as a survey of recent results linking super-Brownian to scaling limits in statistical mechanics.Comment: 34 page

Spread-out percolation in R^d

Let $X$ be either $Z^d$ or the points of a Poisson process in $R^d$ of intensity 1. Given parameters $r$ and $p$, join each pair of points of $X$ within distance $r$ independently with probability $p$. This is the simplest case of a `spread-out' percolation model studied by Penrose, who showed that, as $r\to\infty$, the average degree of the corresponding random graph at the percolation threshold tends to 1, i.e., the percolation threshold and the threshold for criticality of the naturally associated branching process approach one another. Here we show that this result follows immediately from of a general result of the authors on inhomogeneous random graphs.Comment: 9 pages. Title changed. Minor changes to text, including updated references to [3]. To appear in Random Structures and Algorithm

Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Renyi random graphs. In this updated version we incorporate an erratum to be published in a forthcoming issue of Probab. Theory Relat. Fields. This results in a modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming issue of Probab. Theory Relat. Field

Competition between growths governed by Bernoulli Percolation

We study a competition model on $\mathbb{Z}^d$ where the two infections are driven by supercritical Bernoulli percolations with distinct parameters $p$ and $q$. We prove that, for any $q$, there exist at most countably many values of $p<\min(q, \overrightarrow{p\_c})$ such that coexistence can occur.Comment: 30 pages with figure

Super-exponential extinction time of the contact process on random geometric graphs

In this paper, we prove lower and upper bounds for the extinction time of the contact process on random geometric graphs with connecting radius tending to infinity. We obtain that for any infection rate $\lambda >0$, the contact process on these graphs survives a time super-exponential in the number of vertices.Comment: Accepted for publication in Combinatorics, Probability and Computin

Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions

We consider self-avoiding walk and percolation in \Zd, oriented percolation in \Zd\times\Zp, and the contact process in \Zd, with $p D(\cdot)$ being the coupling function whose range is denoted by $L<\infty$. For percolation, for example, each bond $\{x,y\}$ is occupied with probability $p D(y-x)$. The above models are known to exhibit a phase transition when the parameter $p$ varies around a model-dependent critical point \pc. We investigate the value of \pc when $d>6$ for percolation and $d>4$ for the other models, and $L\gg1$. We prove in a unified way that \pc=1+C(D)+O(L^{-2d}), where the universal term 1 is the mean-field critical value, and the model-dependent term $C(D)=O(L^{-d})$ is written explicitly in terms of the function $D$. Our proof is based on the lace expansion for each of these models.Comment: 22 pages, no figure

The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^{-3/2}, plus an error term of order n^{-3/2-\epsilon} with \epsilon >0. This is a strong statement that the critical exponent delta is given by delta =2.Comment: 56 pages, 3 Postscript figures, in AMS-LaTeX, with graphicx, epic, and xr package