Piotr Sniady (IMPAN, Warsaw): Exact formulas for coalescing particle systems

Consider particles on the integers or the real line that perform random walks and coalesce when they collide. The classical Karlin–McGregor theorem gives determinantal formulas for non-colliding particles on the line, but coalescence reduces the particle count and breaks the square matrix structure. Recently, Ákos Urbán extended the determinant to coalescing Pólya walks. We take a different approach: at each collision, a ghost particle continues along an independent path, preserving the total particle count and yielding determinantal formulas for any skip-free process at once. Summing out ghost positions recovers Urbán's formula. Applications include Rayleigh gap distributions, a Pfaffian point process structure for basin boundaries (extending Tribe–Zaboronski and Garrod–Poplavskyi–Tribe–Zaboronski), adaptation to annihilating systems, and a central limit theorem.