In the bootstrap percolation model, sites in an L by L square are initially independently declared active with probability p. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as p → 0 and L → ∞ simultaneously of the probability I (L, p) that the entire square is eventually active. We prove that I (L, p) → 1 if lim inf p log L > λ, and I (L, p) → 0 if lim sup p log L < λ, where λ = π 2 /18. We prove the same behaviour, with the same threshold λ, for the probability J (L, p) that a site is active by time L in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold λ = π 2 /6. The existence of the thresholds λ, λ settles a conjecture of Aizenman and Lebowitz [3], while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony [2].

Abstract. We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

Let \Pi be an ergodic simple point process on R^d and let \Pi^* be its Palm version. Thorisson [Ann. Probab. 24 (1996) 2057-2064] proved that there exists a shift coupling of \Pi and \Pi^*; that is, one can select a (random) point Y of \Pi such that translating \Pi by -Y yields a configuration whose law is that of \Pi^*. We construct shift couplings in which Y and \Pi^* are functions of \Pi, and prove that there is no shift coupling in which \Pi is a function of \Pi^*. The key ingredient is a deterministic translation-invariant rule to allocate sets of equal volume (forming a partition of R^d) to the points of \Pi. The construction is based on the Gale-Shapley stable marriage algorithm [Amer. Math. Monthly 69 (1962) 9-15]. Next, let \Gamma be an ergodic random element of {0,1}^{Z^d} and let \Gamma^* be \Gamma conditioned on \Gamma(0)=1. A shift coupling X of \Gamma and \Gamma^* is called an extra head scheme. We show that there exists an extra head scheme which is a function of \Gamma if and only if the marginal E[\Gamma(0)] is the reciprocal of an integer. When the law of \Gamma is product measure and d\geq3, we prove that there exists an extra head scheme X satisfying E\exp c\|X\|^d<\infty; this answers a question of Holroyd and Liggett [Ann. Probab. 29 (2001) 1405-1425].Comment: Published at http://dx.doi.org/10.1214/009117904000000603 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A sorting network is a shortest path from 12 · · · n to n · · · 21 in the Cayley graph of S n generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n → ∞ the spacetime process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Hölder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux. τ s 1 τ s 2 · · · τ s N = ρ

Suppose that red and blue points occur as independent homogeneous Poisson processes in R d . We investigate translation-invariant schemes for perfectly matching the red points to the blue points. For any such scheme in dimensions d = 1, 2, the matching distance X from a typical point to its partner must have infinite d/2-th moment, while in dimensions d ≥ 3 there exist schemes where X has finite exponential moments. The Gale-Shapley stable marriage is one natural matching scheme, obtained by iteratively matching mutually closest pairs. A principal result of this paper is a power law upper bound on the matching distance X for this scheme. A power law lower bound holds also. In particular, stable marriage is close to optimal (in tail behavior) in d = 1, but far from optimal in d ≥ 3. For the problem of matching Poisson points of a single color to each other, in d = 1 there exist schemes where X has finite exponential moments, but if we insist that the matching is a deterministic factor of the point process then X must have infinite mean.