Percolation in High Dimensions

Critical percolation in high dimensions: critical exponents, finite size scaling and random walks

The Time of Bootstrap Percolation in Two Dimensions

We study the distribution of the percolation time T of 2-neighbour bootstrap percolation on [n] with initial set A ∼ Bin([n], p). We determine T up to a constant factor with high probability for all p above the critical probability for percolation, and to within a 1 + o(1) factor for a large range of p.

The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents

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 η and δ, for the nearest-neighbour model in very high dimensions d 6 and for sufficiently spreadout models in all dimensions d > 6. The exponent η describes the low frequency behaviour of the Fourier transform of the cri...

The Incipient Infinite Cluster in High-dimensional Percolation

We announce our recent proof that, for independent bond percolation in high dimensions, the scaling limits of the incipient infinite cluster’s two-point and three-point functions are those of integrated super-Brownian excursion (ISE). The proof uses an extension of the lace expansion for percolation.

THE SCALING LAW FOR THE DISCRETE KINETIC GROWTH PERCOLATION MODEL

The Scaling Law for the Discrete Kinetic Growth Percolation Model The critical exponent of the total number of finite clusters α is calculated directly without using scaling hypothesis both below and above the percolation threshold pc based on a kinetic growth percolation model in two and three dimensions. Simultaneously, we can calculate other critical exponents β and γ, and show that the scal...

Some remarks on AB-percolation in high dimensions

In this paper we consider the AB-percolation model on Z1 d and Z. Let pH (Zd) be the critical probability for AB-percolation on Z. We show that pH (Zd) ;1/(2d). If the probability of a site to be in state A is g/(2d) for some fixed g.1, then the probability that AB-percolation occurs converges as d→` to the unique strictly positive solution y(g) of the equation y512exp(2gy). We also find the li...