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.
From: Piotr Śniady [view email]
[v1]
Wed, 11 Feb 2026 12:16:16 UTC (47 KB)
[v2]
Mon, 9 Mar 2026 13:48:28 UTC (47 KB)
[v3]
Wed, 8 Jul 2026 16:21:03 UTC (81 KB)
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