Using a recently developed method to simulate percolation on large clusters of distributed machines [1], we have numerically calculated crossing, spanning and wrapping probabilities in two-dimensional site and bond percolation with exceptional accuracy. Our results are fully consistent with predictions from Conformal Field Theory. We present many new results that await theoretical explanation, ...

We study the percolation in the hierarchical lattice of order N where the probability of connection between two nodes separated by a distance k is of the form min{αβ−k,1}, α ≥ 0 and β > 0. We focus on the vertex degrees of the resulting percolation graph and on whether there exists an infinite component. For fixed β , we show that the critical percolation value αc(β ) is non-trivial, i.e., αc(β...

The probability that the cluster of the origin in critical site percolation on the triangular grid has diameter larger than R is proved to decay like R−5/48 as R→ ∞.

Consider independent bond percolation with retention probability p on a spherically symmetric tree Γ. Write θΓ(p) for the probability that the root is in an infinite open cluster, and define the critical value pc = inf{p : θΓ(p) > 0}. If θΓ(pc) = 0, then the root may still percolate in the corresponding dynamical percolation process at the critical value pc, as demonstrated recently by Häggströ...

We discuss inequalities and applications for percolation and randomcluster models. The relevant areas of methodology concern the following two types of inequality: inequalities involving the probability of a general increasing event, and certain differential inequalities involving the percolation probability. We summarise three areas of application of such inequalities, namely strict inequality...

A new substitution method improves bounds for critical probabilities of the bond percolation problem on the Kagomé lattice, K. The method theoretically produces a sequence of upper and lower bounds, in which the second pair of bounds establish .5182 ≤ pc(K) ≤ .5335.

We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy’s formula. Then we prove existence, uniqueness, and conformal invarianc...

The magnetic and electrical properties of Co(AsxS1-~)2 are investigated and the phase diagrams are shown. In CO(AS~SI-~)Z, the outbreak concentration of the ferromagnetism is obtained to be xc = 0.37 + 0.01, while the critical concentration of the metal-non metal transition is determined to be X ; = 0.47 f 0.01. The results obtained are explained by site and bond percolation mechanisms in the d...

We investigate a recent suggestion that the spatial distribution of earthquake hypocenters makes a fractal set with a structure and fractal dimensionality close to those of the backbone of critical percolation clusters, by analyzing four different sets of data for the hypocenter distributions and calculating the dynamical properties of the geometrical distribution such as the spectral dimension...