Karlin-McGregor formula for coalescing random walks
When particles performing random walks on a line collide, they may coalesce (merge into one) or annihilate (both are destroyed). Computing exact probabilities for such systems has been difficult because collisions change the particle count, while classical determinantal methods (the Karlin-McGregor / Lindström-Gessel-Viennot formula) require a fixed number of particles throughout. This series of papers introduces ghost particles — invisible walkers that replace the missing particles at each collision — restoring the square matrix structure needed for determinantal and Pfaffian formulas. The results are exact, hold for arbitrary nearest-neighbor random walks and their Brownian scaling limits, and connect to Pfaffian point process theory.
Relationships between papers
constructions
Preprints
C1: Exact determinant formulas for coalescing particle systems
Piotr Śniady. Exact determinant formulas for coalescing particle systems.
arXiv:2602.10782 |
experimental pre-release version
Introduces ghost particles: at each collision, one particle emerges as usual and one invisible ghost continues alongside it, preserving the total count. This restores the square matrix structure needed for a determinantal formula. The probability of any specified coalescence pattern is given by a determinant whose entries are single-particle transition probabilities. Integrating out ghost positions yields a closed-form formula for the surviving particles alone: the coalescence determinant.
C2: Coalescing random walks via the coalescence determinant
Piotr Śniady. Coalescing random walks via the coalescence determinant.
arXiv:2602.20043 |
experimental pre-release version
Applies the coalescence determinant to the wall-particle system: when every site is initially occupied, this is the joint system of survivors and the boundaries between their basins of attraction. Its finite-dimensional distributions are determinants of block matrices built from transition probabilities and their cumulative sums. As applications, recovers the Rayleigh spacing density and the joint distribution of consecutive gaps by new methods.
A1: Determinant and Pfaffian formulas for particle annihilation
Piotr Śniady. Determinant and Pfaffian formulas for particle annihilation.
arXiv:2602.13183 |
experimental pre-release version
Adapts the ghost particle method to annihilation, where both colliding particles are destroyed. Both trajectories continue as invisible ghosts, yielding an exact determinantal formula. For complete annihilation (no survivors), the formula simplifies to a Pfaffian. Since pairwise coalescence can be reinterpreted as complete annihilation, this also yields a Pfaffian coalescence formula.
C3: Pfaffian structure of basin walls for coalescing particles
Piotr Śniady. Pfaffian structure of basin walls for coalescing particles.
arXiv:2602.22885 |
experimental pre-release version
Returns to coalescence: when every site is initially occupied by a coalescing particle, the walls between basins of attraction form a Pfaffian point process. The key tool is a checkerboard duality that identifies the walls of the coalescing system with surviving particles of an annihilating system, so the Pfaffian formulas from A1 apply directly. This yields an exact Pfaffian empty-interval formula for the walls and explicit cumulants of wall indicators. A structural property of these cumulants, indecomposability, implies a central limit theorem for the wall count.