Pfaffian structure of basin walls for coalescing particles

Abstract

Coalescing particles on a line merge when they meet. As they do, their basins of attraction merge and the walls between basins disappear. If every site is initially occupied, these walls at any positive time form a Pfaffian point process: all correlation functions are determined by pairwise quantities arranged in antisymmetric matrices. Tribe, Zaboronski, Garrod, and Poplavskyi established this structure using analytic methods for time-homogeneous dynamics; our combinatorial approach works for any skip-free process (one where particles cannot change order without first meeting). We show that the Pfaffian structure lives naturally at the wall level: we prove an exact Pfaffian empty-interval formula for the walls and compute the cumulants of the wall indicators (higher-order analogs of the variance) as signed sums of probabilities that independent particles started at the interval endpoint positions reorder in specific ways. A structural property of these sums, indecomposability - every nonzero term couples all wall positions together - implies a central limit theorem for the wall count. A checkerboard duality identifies the walls of one process with the surviving particles of the dual process. This covers totally asymmetric dynamics and position-dependent transition rules, and for Brownian motion recovers the known Pfaffian point process.