Restricted Percolation Critical Exponents in High Dimensions

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

Critical exponents in percolation via lattice animals

We examine the percolation model by an approach involving lattice animals, divided according to their surface-area-to-volume ratio. Throughout, we work with the bond percolation model in Z. However, the results apply to the site or bond model on any infinite transitive amenable graph with inessential changes. For any given p ∈ (0, 1), two lattice animals with given size are equally likely to ar...

Percolation critical exponents in scale-free networks.

We study the behavior of scale-free networks, having connectivity distribution P(k) approximately k(-lambda), close to the percolation threshold. We show that for networks with 3<lambda<4, known to undergo a transition at a finite threshold of dilution, the critical exponents are different than the expected mean-field values of regular percolation in infinite dimensions. Networks with 2<lambda<...

Critical exponents for two-dimensional percolation

We show how to combine Kesten’s scaling relations, the determination of critical exponents associated to the stochastic Loewner evolution process by Lawler, Schramm, and Werner, and Smirnov’s proof of Cardy’s formula, in order to determine the existence and value of critical exponents associated to percolation on the triangular lattice.

Percolation in High Dimensions

Letpc{d) be the critical probability for percolation in Z . It is shown that lim,, _, ro 2dpc(d) = 1. The proof uses the properties of a random subgraph of an m-ary d-dimensional cube. If each edge in this cube is included with probability greater than \/2d{\ — 3/m), then, for large d, the cube will have a connected component of size cm for some c> 0. This generalizes a result of Ajtai, Komlds ...