Supercritical causal maps: geodesics and simple random walk

Scaling limit of simple random walk on supercritical percolation clusters

Cut times for Simple Random Walk Cut times for Simple Random Walk

Let S(n) be a simple random walk taking values in Z d. A time n is called a cut time if S0; n] \ Sn + 1; 1) = ;: We show that in three dimensions the number of cut times less than n grows like n 1? where = d is the intersection exponent. As part of the proof we show that in two or three dimensions PfS0; n] \ Sn + 1; 2n] = ;g n ? ; where denotes that each side is bounded by a constant times the ...

Scaling of a random walk on a supercritical contact process

A proof is provided of a strong law of large numbers for a one-dimensional random walk in a dynamic random environment given by a supercritical contact process in equilibrium. The proof is based on a coupling argument that traces the space-time cones containing the infection clusters generated by single infections and uses that the random walk eventually gets trapped inside the union of these c...

Favourite sites of simple random walk

We survey the current status of the list of questions related to the favourite (or: most visited) sites of simple random walk on Z, raised by Pál Erdős and Pál Révész in the early eighties.

The Intersection Exponent For Simple Random Walk

The intersection exponent for simple random walk in two and three dimensions gives a measure of the rate of decay of the probability that paths do not intersect. In this paper we show that the intersection exponent for random walks is the same as that for Brownian motion and show in fact that the probability of nonintersection up to distance n is comparable (equal up to multiplicative constants...