Determinant and Pfaffian formulas for particle annihilation

TL;DR

This work develops exact finite-configurations for one-dimensional annihilating random walks by introducing ghost particles that preserve a fixed entity count, enabling determinantal formulas for annihilation outcomes. The central result is an annihilation formula expressed as a determinant with formal variables that filter to the physically admissible ghost configurations; a Pfaffian form emerges in the complete-annihilation limit, linking to Pfaffian point process theory. The authors also establish a deep connection between annihilation and coalescence through a parity-based reclassification, deriving a Pfaffian formula for pairwise coalescence and highlighting a broader determinant–Pfaffian spectrum across models. The framework applies to discrete lattice paths, birth–death chains, and Brownian motion, and it offers exact, finite-time probabilities for domain-wall dynamics and reaction–diffusion systems, with companion papers extending the approach to gaps and continuous-time settings.

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.

Figures (6)

• Figure 1: Annihilation on the checkerboard lattice. Four particles start at $x_1 < x_2 < x_3 < x_4$. Particles $2$ (double) and $3$ (zigzag) annihilate at $c$; both are destroyed and an ordered pair of ghosts emerges (dashed paths). Particles $1$ (solid) and $4$ (tick marks) survive. Ghost paths freely cross survivor paths (shown offset)---ghosts do not interact. Final positions: $a < y_1 < b < y_2$.
• Figure 2: An annihilation performance on the lattice $\mathbb{Z}^2$ with North/East steps. Five particles start at $x_1, \ldots, x_5$. Particles $2$ and $3$ meet at $c_1$; both are destroyed and ghost pair $1$ emerges (dashed paths). Particles $1$ and $4$ meet at $c_2$; both are destroyed and ghost pair $2$ emerges (dotted paths). Particle $5$ survives, reaching $s_1$. Within each ghost pair, one ghost follows a single line, the other a double line. Ghost pairs are distinguished by line pattern (dashed vs. dotted)---they carry no memory of which particles were destroyed.
• Figure 3: Two-particle annihilation, case $\varepsilon_j = +1$: $a_j \preceq b_j$. (a) Schema: particles $I$ (thick) and $J$ (wavy) collide; both are destroyed and two ghost paths emerge (dashed). Four distinct styles emphasize that no identity persists through the collision. (b) Attribution via the swap principle: particle $I$ (left, thick) is glued to the rightward ghost at $b_j$ (thick dashed); particle $J$ (right, wavy) is glued to the leftward ghost at $a_j$ (wavy dashed). The spatial ordering is reversed.
• Figure 4: Successful and failed castings (annihilation, $n=4$, $k=1$). Same final state as \ref{['fig:intro-annihilation']}: survivors at $y_1$, $y_2$ and ghost pair at $a$, $b$. (a) The bijection $\pi$ assigns particles $2$ and $3$ to the ghost pair, particles $1$ and $4$ to survivors. At the collision point $c$, particles $2$ and $3$ are destined for the same ghost pair---a valid annihilation. Both become ghosts after the collision (dashed). (b) The bijection assigns particles $1$ and $3$ to the ghost pair, particles $2$ and $4$ to survivors. At their first crossing, particles $1$ and $2$ meet, but particle $2$ is destined for a survivor position---the crossing is spurious (red marker). Rehearsal fails.
• Figure 5: The segment swap operation. (a) Paths $P_1$ (solid) and $P_2$ (double) cross at vertex $c$. (b) After the swap, final segments are exchanged: $P'_1$ follows $P_1$ to $c$, then $P_2$'s tail to $y_2$; $P'_2$ follows $P_2$ to $c$, then $P_1$'s tail to $y_1$. The paths still cross at $c$, but now go to swapped endpoints.

Theorems & Definitions (46)

• Theorem 1: Annihilation formula with ghosts
• Definition 2.1: Spacetime graph
• Definition 2.2: Paths and weights
• Definition 2.3: Source and target sets
• Definition 2.4: Planar configuration
• Definition 2.5: Actors
• Definition 2.6: Roles
• Definition 2.7: Collision diagram
• Definition 2.8: Performance
• Definition 2.9: Ghost sign