We study the ( )-color cyclic particle system on the one-dimensional integer lattice ( Z ), first introduced by Bramson and Griffeath (Ann Prob:26–45, 1989). In that paper they show that almost surely, every site changes its color infinitely often if ( 3,4 ) and only finitely many times if ( 5 ). In addition, they conjecture that for ( 3,4 ) the system clusters, that is, for any pair of sites x, y, with probability tending to 1 as (t ), x and y have the same color at time t. Here we prove that conjecture.
We study the kinetics of ballistic annihilation, A+A→0, with continuous initial particle velocity distributions. The concentration and the rms velocity are found to decay as c∼t -α and υ rms ∼t -β , respectively, with the relation α+β=1 holding in any spatial dimension. A «mean-field» Boltzmann equation for the evolution of the velocity distribution predicts that a and β depend strongly on the initial condition, a behavior which is confirmed numerically in one and two dimensions  Bullets are fired, one per second, with independent speeds sampled uniformly from a discrete set. Collisions result in mutual annihilation. We show that the second fastest bullet survives with positive probability, while a slowest bullet does not. This also holds for exponential spacings between firing times, and for certain non-uniform measures that place less probability on the second fastest bullet. Our results provide new insights into a two-sided version of the bullet process known to physicists as ballistic annihilation.