We argue that comparison with observations of theoretical models for the velocity distribution of pulsars must be done directly with the observed quantities, i.e. parallax and the two components of proper motion. We develop a formalism to do so, and apply it to pulsars with accurate VLBI measurements. We find that a distribution with two maxwellians improves significantly on a single maxwellian. The 'mixed' model takes into account that pulsars move away from their place of birth, a narrow region around the galactic plane. The best model has 42% of the pulsars in a maxwellian with average velocity σ √ 8/π = 120 km/s, and 58% in a maxwellian with average velocity 540 km/s. About 5% of the pulsars has a velocity at birth less than 60 km/s. For the youngest pulsars (τ c < 10 Myr), these numbers are 32% with 130 km/s, 68% with 520 km/s, and 3%, with appreciable uncertainties.

We study the last-passage growth model on the planar integer lattice with exponential weights. With boundary conditions that represent the equilibrium exclusion process as seen from a particle right after its jump we prove that the variance of the last-passage time in a characteristic direction is of order t 2/3 . With more general boundary conditions that include the rarefaction fan case we show that the last-passage time fluctuations are still of order t 1/3 , and also that the transversal fluctuations of the maximal path have order t 2/3 . We adapt and then build on a recent study of Hammersley's process by Cator and Groeneboom, and also utilize the competition interface introduced by Ferrari, Martin and Pimentel. The arguments are entirely probabilistic, and no use is made of the combinatorics of Young tableaux or methods of asymptotic analysis.

Since the Susceptible-Infected-Susceptible (SIS) epidemic threshold is not precisely defined in spite of its practical importance, the classical SIS epidemic process has been generalized to the ε−SIS model, where a node possesses a self-infection rate ε, in addition to a link infection rate β and a curing rate δ. The exact Markov equations are derived, from which the steady state can be computed. The major advantage of the ε−SIS model is that its steady state is different from the absorbing (or overall-healthy state) and approximates, for a certain range of small ε > 0, the in reality observed phase transition, also called the "metastable" state, that is characterized by the epidemic threshold. The exact steady-state analysis for the complete graph illustrates the effect of small ε and the quality of the first-order mean-field approximation, the N -intertwined model, proposed earlier. Apart from duality principles, often used in the mathematical literature, we present an exact recursion relation for the Markov infinitesimal generator.

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We show that, for a stationary version of Hammersley's process, with Poisson "sources" on the positive x-axis, and Poisson "sinks" on the positive y-axis, an isolated second-class particle, located at the origin at time zero, moves asymptotically, with probability 1, along the characteristic of a conservation equation for Hammersley's process. This allows us to show that Hammersley's process without sinks or sources, as defined by Aldous and Diaconis [Probab. Theory Related Fields 10 (1995) 199-213] converges locally in distribution to a Poisson process, a result first proved in Aldous and Diaconis (1995) by using the ergodic decomposition theorem and a construction of Hammersley's process as a one-dimensional point process, developing as a function of (continuous) time on the whole real line. As a corollary we get the result that EL(t, t)/t converges to 2, as t → ∞, where L(t, t) is the length of a longest North-East path from (0, 0) to (t, t). The proofs of these facts need neither the ergodic decomposition theorem nor the subadditive ergodic theorem. We also prove a version of Burke's theorem for the stationary process with sources and sinks and briefly discuss the relation of these results with the theory of longest increasing subsequences of random permutations.