Determinant and Pfaffian formulas for particle annihilation

We consider systems of particles on a line in which colliding particles annihilate each other and vanish.

Computing exact annihilation probabilities is difficult because every collision reduces the particle count, while determinantal methods require a fixed count throughout.

The ghost particle method, introduced in a companion paper for coalescence, removes the obstacle: destroyed particles continue walking as invisible ghosts, so the number of trajectories never changes.

Applied to annihilation, the method yields an exact determinantal formula for the probability of any prescribed outcome - the number of annihilations, the survivor positions, and the positions of the ghosts.

For complete annihilation, where no particle survives, the determinant collapses to a Pfaffian, an algebraic relative of the determinant built from pairwise quantities: although the particles interact, the extinction probability is determined by pairwise annihilation probabilities alone.

This gives a combinatorial explanation of the Pfaffian structure of annihilating systems, previously derived through differential equations for specific dynamics.

The annihilation formula also yields results about coalescence: the event that prescribed pairs of particles have merged can be reinterpreted as complete annihilation, producing a Pfaffian coalescence formula.

All formulas are exact for any finite initial configuration and apply to discrete lattice paths, birth-death chains, and continuous diffusions including Brownian motion.