Exact determinant formulas for coalescing particle systems

TL;DR

The paper introduces ghost particles to preserve a fixed particle count in coalescing systems, enabling exact determinant formulas for prescribed coalescence patterns. It generalizes the Karlin–McGregor/LGV framework to interacting particles by encoding collisions with ghosts and using a sign-reversing involution to cancel spurious contributions. The main result is a coalescence determinant with formal ghost variables, plus a ghost-free variant that yields the distribution of surviving heir positions. The approach applies across discrete lattice paths, birth–death chains, and continuous diffusions, and connects to voter models and population genetics while enabling extensions to annihilation and gap statistics in companion work. Overall, the method provides a unified, combinatorial, and exact toolkit for finite-time coalescence probabilities in a broad class of stochastic processes.

Abstract

When particles on a line collide, they may coalesce into one. Such systems model diverse phenomena, from voter dynamics and opinion formation to ancestral lineages in population genetics. Computing exact coalescence probabilities has been difficult because collisions reduce the particle count, while classical determinantal methods require a fixed count throughout. We introduce ghost particles: when two particles merge, the discarded trajectory continues as an invisible walker. Ghosts restore the missing dimension, enabling an exact determinantal formula. We prove that the probability of any specified coalescence pattern is given by a determinant. Integrating out ghost positions yields a closed-form ghost-free formula for the surviving particles. The framework applies to discrete lattice paths, birth-death chains, and continuous diffusions including Brownian motion.

Figures (11)

• Figure 1: Coalescence on the checkerboard lattice (schema $2+1$). Three particles start at $x_1 < x_2 < x_3$. Particles $1$ (solid) and $2$ (double) coalesce at $c$, producing heir $1$ (ticked) and a ghost (dotted). Particle $3$ (zigzag) does not coalesce; it is heir $2$. The ghost shares two edges with heir $2$ (shown offset)---ghosts do not interact and may cross any path freely. Final positions: $y_1 < y_2 < z$.
• Figure 2: The coalescence matrix: staircase structure. Matrix $M$ for the $2{+}2$ coalescence schema (four initial particles; two heirs, two ghosts). Heir columns (yellow) contain transition probabilities $p(x_i \to H_j)$. Ghost columns show the staircase pattern: entries with row index $i < \mathop{\mathrm{rank}}\nolimits(g)$ (blue, solid lines) have $-p \cdot t^-$; entries with $i \geq \mathop{\mathrm{rank}}\nolimits(g)$ (orange, dashed lines) have $p \cdot t^+$. The thick staircase line separates the two regions. Coefficient extraction $[\prod_g t_g^{\varepsilon(g)}]$ selects terms consistent with the ghost signs.
• Figure 3: Running example: a coalescence performance. A performance records the collision structure on a spacetime graph---here the lattice $\mathbb{Z}^2$ with North/East steps. Four particles start at $x_{I_1}$, $x_{I_2}$, $x_{I_3}$, $x_{I_4}$ (leaves, colored dots). Particles $I_2$ and $I_3$ meet at $z_3$, merging into $[2,4]$; then $I_1$ and $[2,4]$ meet at $z_2$, forming heir $H=[1,4]$ which ends at the root $y_{[1,4]}$. Particle $I_4$ does not coalesce; it is heir $H'=[4,5]$. Ghost paths (dashed/dotted) emanate from merger points: ghost $2$ is born at $z_2$, ghost $3$ at $z_3$. Note that ghost $3$ crosses through particle $I_1$---ghosts are non-interacting. See \ref{['fig:boundary']} for the labeling scheme.
• Figure 4: Interval labeling and the final state. Particles are labeled by unit intervals; final entities by intervals (heirs) or junction points (ghosts). This diagram shows the same example as \ref{['fig:genealogy']}. Here $n=4$ (\ref{['ex:final-state']}): initial particles $\mathcal{A} = \{I_1, I_2, I_3, I_4\}$ and final entities $\mathcal{R} = \{H, 2, 3, H'\}$. Heirs $H = [1,4]$ and $H' = [4,5]$; ghosts appear at junctions $2$ and $3$. The $\min$ function is indicated by the small dots: each interval $[a,b]$ has a dot at position $a$ (its left endpoint), so $\min$ reads off the horizontal coordinate of the dot. Arrows show one bijection $\pi$: $I_1 \mapsto H$, $I_2 \mapsto 3$, $I_3 \mapsto H'$, $I_4 \mapsto 2$. Under $\min$, this becomes the permutation $1 \mapsto 1$, $2 \mapsto 3$, $3 \mapsto 4$, $4 \mapsto 2$---the $3$-cycle $(2\ 3\ 4)$ with sign $+1$.
• Figure 5: The consecutive collision property. A forbidden configuration: paths $P_1$ and $P_3$ meet at $v$ and continue as a trunk. Path $P_2$ attempts to "tunnel" around $P_3$ without crossing, then hit the trunk at $w$. \ref{['itm:consecutive']} in \ref{['def:planar']} forbids this: $P_2$ must cross $P_1$ or $P_3$ before $v$.

Theorems & Definitions (58)

• Theorem 1: Coalescence formula
• Definition 2.1: Spacetime graph
• Definition 2.2: Paths and weights
• Example 1: Main examples
• Definition 2.3: Source and target sets
• Definition 2.4: Planar configuration
• Definition 2.5: Interval labeling
• Definition 2.6: Heir function
• Example 2: Final state for $n = 4$
• Definition 2.7: Rank