Determinant and Pfaffian formulas for particle annihilation

Abstract

When particles on a line collide, they may annihilate - both are destroyed. Computing exact annihilation probabilities has been difficult because collisions reduce the particle count, while determinantal methods require a fixed count throughout. The ghost particle method, introduced in a companion paper for coalescence, keeps destroyed particles walking as invisible ghosts that restore the missing dimension. We apply this method to annihilation: when two particles annihilate, both trajectories continue as invisible walkers, yielding an exact determinantal formula that specifies the number of annihilations, where survivors end up, and where ghosts end up. For complete annihilation (no survivors), the formula simplifies to a Pfaffian - an algebraic relative of the determinant built from pairwise quantities - connecting to Pfaffian point process theory. The annihilation formula also yields results about coalescence: pairwise coalescence can be reinterpreted as complete annihilation, producing a Pfaffian coalescence formula. These formulas are exact for any finite initial configuration and apply to discrete lattice paths, birth-death chains, and continuous diffusions including Brownian motion.