Second class particles and cube root asymptotics for Hammersley's process

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, the variance of the length of a longest weakly North--East path $L(t,t)$ from $(0,0)$ to $(t,t)$ is equal to $2\mathbb {E}(t-X(t))_+$, where $X(t)$ is the location of a second class particle at time $t$. This implies that both $\mathbb {E}(t-X(t))_+$ and the variance of $L(t,t)$ are of order $t^{2/3}$. Proofs are based on the relation between the flux and the path of a second class particle, continuing the approach of Cator and Groeneboom [Ann. Probab. 33 (2005) 879--903].Comment: Published at http://dx.doi.org/10.1214/009117906000000089 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A new look at distances and velocities of neutron stars

We take a fresh look at the determination of distances and velocities of neutron stars. The conversion of a parallax measurement into a distance, or distance probability distribution, has led to a debate quite similar to the one involving Cepheids, centering on the question whether priors can be used when discussing a single system. With the example of PSRJ0218+4232 we show that a prior is necessary to determine the probability distribution for the distance. The distance of this pulsar implies a gamma-ray luminosity larger than 10% of its spindown luminosity. For velocities the debate is whether a single Maxwellian describes the distribution for young pulsars. By limiting our discussion to accurate (VLBI) measurements we argue that a description with two Maxwellians, with distribution parameters sigma1=77 and sigma2=320 km/s, is significantly better. Corrections for galactic rotation, to derive velocities with respect to the local standards of rest, are insignificant.Comment: Has appeared in Journal of Astrophysics and Astronomy special issue on 'Physics of Neutron Stars and Related Objects', celebrating the 75th birth-year of G. Srinivasan. Ten pages, nine figure

Hammersley's process with sources and sinks

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\to\infty, 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.Comment: Published at http://dx.doi.org/10.1214/009117905000000053 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Space-time stationary solutions for the Burgers equation

We construct space-time stationary solutions of the 1D Burgers equation with random forcing in the absence of periodicity or any other compactness assumptions. More precisely, for the forcing given by a homogeneous Poissonian point field in space-time we prove that there is a unique global solution with any prescribed average velocity. These global solutions serve as one-point random attractors for the infinite-dimensional dynamical system associated to solutions to the Cauchy problem. The probability distribution of the global solutions defines a stationary distribution for the corresponding Markov process. We describe a broad class of initial Cauchy data for which the distribution of the Markov process converges to the above stationary distribution. Our construction of the global solutions is based on a study of the field of action-minimizing curves. We prove that for an arbitrary value of the average velocity, with probability 1 there exists a unique field of action-minimizing curves initiated at all of the Poissonian points. Moreover action-minimizing curves corresponding to different starting points merge with each other in finite time.Comment: 50 pages. In this version: small technical corrections in Lemmas 6.1 and 6.

Cube root fluctuations for the corner growth model associated to the exclusion process

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.Comment: 41 pages, 4 figure

The observed velocity distribution of young pulsars

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 sigma sqrt{8/pi}=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 (tau_c<10 Myr), these numbers are 32% with 130 km/s, 68% with 520 km/s, and 3%, with appreciable uncertainties.Comment: submitted to A&A; 14 pages, 10 figure