An analytic approach is introduced for the determination of rigorous lower bounds for the critical probability of bond percolation in an oriented lattice. This is illustrated by an example, the oriented square lattice in two dimensions.

In this note, we prove that the contact process on the one-dimensional longrange percolation with high exponent exhibits a non-trivial phase transition: there is a critical value of the infection rate λc > 0, such that if λ > λc, the contact process survives with positive probability, whereas if λ < λc, it dies out a.s.

First, we will briefly introduce oriented percolation with our focus on a special orientation on the square grid. Then we will discuss critical probabilities of bond or site percolation on this particular orientation. Finally, we will introduce the Fire-fighting Problem and prove a theorem regarding an upper bound for these critical probabilities in the case that fractional fire-fighters were a...

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.

s: For beginners: This review tries to explain percolation through the cluster properties; it can also be used as an introduction to critical phenomena at other phase transitions for readers not familiar with scaling theory. In percolation each site of a periodic lattice is randomly occupied with probability p or empty with probability I — p. An s-cluster is agroup of s occupied sites connected...

We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly ...

The following full text is a preprint version which may differ from the publisher's version. Abstract: We study the critical probability p c (M) in two-dimensional M-adic fractal percolation. To find lower bounds, we compare fractal perco-lation with site percolation. Fundamentally new is the construction of an computable increasing sequence that converges to p c (M). We prove that p c (2) > 0....

We consider biased random walk on supercritical percolation clusters in Z. We show that the random walk is transient and that there are two speed regimes: If the bias is large enough, the random walk has speed zero, while if the bias is small enough, the speed of the random walk is positive.

A generalized model of percolation encompassing both the usual model, in which bonds are occupied with probability p and are vacant with probability (1−p), and the model appropriate to the statistics of lattice animals, in which the fugacity for occupied bonds is p and that for unoccupied bonds is unity, is formulated. Within this model we discuss the crossover between the two problems and we s...

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...