Bullets are fired from the origin of the positive real line, one per second, with independent speeds sampled uniformly from a discrete set. Collisions result in mutual annihilation. We show that a bullet with the second largest speed survives with positive probability, while a bullet with the smallest speed does not. This also holds for exponential spacings between firing times. Our results imply that the middle-velocity particle survives with positive probability in a two-sided version of the bullet process with three speeds known to physicists as ballistic annihilation.

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The bullet problem with discrete speeds

Ballistic annihilation is an interacting system in which particles placed throughout the real line move at preassigned velocities and annihilate upon colliding. The longstanding conjecture that in the symmetric three-velocity setting there exists a phase transition for the survival of middle-velocity particles was recently resolved by Haslegrave, Sidoravicius, and Tournier. We develop a framework based on a mass transport principle to analyze three-velocity ballistic annihilation with asymmetric velocities assigned according to an asymmetric probability measure. We show the existence of a phase transition in all cases by deriving universal bounds. In particular, all middle-speed particles perish almost surely if their initial density is less than 1/5, regardless of the velocities, relative densities, and spacing of initial particles. We additionally prove the continuity of several fundamental statistics as the probability measure is varied.

The phase structure of asymmetric ballistic annihilation

We consider ballistic annihilation, a model for chemical reactions first introduced in the 1980's physics literature. In this particle system, initial locations are given by a renewal process on the line, motions are ballistic - i.e. each particle is assigned an i.i.d. constant velocity - and collisions between pairs of particles result in mutual annihilation. 
We focus on the case when the velocities are symmetrically distributed among three values, i.e. particles either remain static (with given probability~$p$) or move at constant velocity uniformly chosen among $\pm1$. We establish that this model goes through a phase transition at $p_c=1/4$ between a subcritical regime where every particle eventually annihilates, and a supercritical regime where a positive density of static particles is never hit, confirming 1990s predictions of Droz et al. for the particular case of a Poisson process. Our result encompasses cases where triple collisions can happen; these are resolved by annihilation of one static and one randomly chosen moving particle. 
Our arguments, of combinatorial nature, show that, although the model is not completely solvable, certain large scale features can be explicitly computed, and are universal, i.e. insensitive to the distribution of the initial point process. In particular, in the critical and subcritical regimes, the asymptotics of the time decay of the densities of each type of particle is universal (among exponentially integrable interdistance distributions) and, in the supercritical regime, the distribution of the ``skyline'' process, i.e. the process restricted to the last particles to ever visit a location, has a universal description. 
We also prove that an alternative model introduced by Burdinski, Gupta and Junge does not share the same universality as our model, and find numerical bounds on its critical probability.

Three-speed ballistic annihilation: phase transition and universality.

We consider a system of annihilating particles where particles start from the points of a Poisson process on the line, move at constant i.i.d. speeds symmetrically distributed in {-1,0,+1} and annihilate upon collision. We prove that particles with speed 0 vanish almost surely if and only if their initial density is smaller than or equal to 1/4, and give an explicit formula for the probability of survival of a stationary particle in the supercritical case, which is in accordance with the predictions of Droz et al. We also observe that the value of this probability is unchanged if the Poisson process of initial locations is replaced by any renewal process with atomless interdistance distribution, and show that this follows from a more general, and unexpected, universality property.

The three-speed ballistic annihilation threshold is 1/4

Ballistic annihilation is an interacting particle system in which particles move throughout the real line at preassigned velocities and mutually annihilate upon colliding. In the symmetric three-velocity setting, we generalize these dynamics to include the possibility that collisions result in coalescence. We provide a conjectural formula for the critical initial density of middle-velocity particles that holds for many cases. One corollary is that there are coalescing systems with no phase transition.

Three-velocity coalescing ballistic annihilation

Three-speed ballistic annihilation starts with infinitely many particles on the real line. Each is independently assigned either speed-$0$ with probability $p$, or speed-$\pm 1$ symmetrically with the remaining probability. All particles simultaneously begin moving at their assigned speeds and mutually annihilate upon colliding. Physicists conjecture when $p \leq p_c = 1/4$ all particles are eventually annihilated. Dygert et. al. prove $p_c \leq .3313$, while Sidoravicius and Tournier describe an approach to prove $p_c \leq .3281$. For the variant in which particles start at the integers, we improve the bound to $.2870$. A renewal property lets us equate survival of a particle to the survival of a Galton-Watson process whose offspring distribution a computer can rigorously approximate. This approach may help answer the nearly thirty-year old conjecture that $p_c >0$.

Shrey Gupta

Papers

The upper threshold in ballistic annihilation

The upper threshold in ballistic annihilation