Coalescing random walks via the coalescence determinant

When identical particles on a line collide, they merge and continue as one. Exact determinantal formulas have long been available for particles conditioned never to collide, but collisions change the number of particles, and exact distributions for the survivors have been obtained only in specific settings and by ad hoc methods. Building on the coalescence determinant introduced in a companion paper, we study the wall-particle system: when every site is initially occupied, this is the joint system of survivors and the boundaries between their basins of attraction. Its finite-dimensional distributions are determinants of block matrices built from transition probabilities and their cumulative sums; a finite block matrix suffices even when the initial configuration is infinite. As applications, we recover the Rayleigh spacing density and the joint distribution of consecutive gaps - which are negatively correlated - by new methods, and give a new derivation of the determinantal formula for the joint CDF of finitely many coalescing particles starting from fixed positions. All formulas hold for arbitrary nearest-neighbor random walks and their Brownian scaling limits, with no specific transition kernels required.