Abstract. We consider random walks with continuous and symmetric step distributions. We prove universal asymptotics for the average proportion of the age of the kth longest lasting record for k = 1, 2, . . . and for the probability that the record of the kth longest age is broken at step n. Due to the relation to the Chinese restaurant process, the ranked sequence of proportions of ages converges to the Poisson-Dirichlet distribution.

We define an inhomogeneous percolation model on "ladder graphs" obtained as direct products of an arbitrary graph G = (V, E) and the set of integers Z (vertices are thought of as having a "vertical" component indexed by an integer). We make two natural choices for the set of edges, producing an unoriented graph G and an oriented graph G. These graphs are endowed with percolation configurations in which independently, edges inside a fixed infinite "column" are open with probability q, and all other edges are open with probability p. For all fixed q one can define the critical percolation threshold p c (q). We show that this function is continuous in (0, 1).2010 Mathematics Subject Classification. MSC 60K35, MSC 82B43.

In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This approach is not only simpler than the original multi-scale renormalisation proof of the result in two and more dimensions, but also gives significantly better bounds. As a byproduct, we improve the best known bounds for the stability threshold of Toom's North-East-Center majority rule cellular automaton.

We give a construction of a tree in which the contact process with any positive infection rate survives but, if a certain privileged edge e * is removed, one obtains two subtrees in which the contact process with infection rate smaller than 1/4 dies out.