Exact determinant formulas for coalescing particle systems

TL;DR

This paper introduces a novel determinantal approach using ghost particles to compute exact coalescence probabilities in particle systems, applicable to various Markov processes including Brownian motion.

Contribution

It develops a new method with ghost particles that preserves matrix structure, enabling exact determinant formulas for coalescence probabilities in Markovian systems.

Findings

Derived explicit determinant formulas for coalescence probabilities.

Applicable to a wide class of Markov processes including diffusions.

Provides closed-form solutions for surviving particles after coalescence.

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: at each collision, one particle emerges as usual and one invisible ghost emerges 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.