Exact determinant formulas for coalescing particle systems

Abstract

When particles on a line collide, they may coalesce into one. Such systems arise in the voter model, where boundaries between opinion clusters perform coalescing random walks, and in reaction-diffusion theory, where diffusing particles merge on contact. Computing exact coalescence probabilities has been difficult because collisions reduce the particle count, while classical determinantal methods require a fixed number of particles throughout. We introduce ghost particles: when two particles collide, one survivor continues as usual and one invisible ghost is created alongside it, preserving the total count. This restores the square matrix structure needed for a determinantal formula. We prove that the probability of any specified coalescence pattern - which initial particles merge into which survivors - is given by a determinant whose entries are transition probabilities. Integrating out ghost positions yields a closed-form formula for the surviving particles alone: the coalescence determinant. The only assumptions are the Markov property and nearest-neighbor transitions, so the results apply wherever the classical non-colliding theory does: discrete lattice paths, birth-death chains, and continuous diffusions including Brownian motion.

56 pages, 16 figures. v3: Ákos Urbán added as author; substantially revised (intrinsic tournament definition of the sign, rewritten performance-casting bijection, explicit construction of the finite coalescing system); discrete results formalized in Lean 4, archived with a Python reference implementation and an extended version at https://doi.org/10.5281/zenodo.21037519