A Karlin-McGregor formula for coalescing random walks

The Karlin-McGregor theorem expresses the probability that $n$ random walks (or Brownian motions) reach specified positions without collision as an $n\times n$ determinant. When particles coalesce upon meeting, this structure appears to break: after $k$ mergers only $n−k$ particles remain, destroying the square matrix. We restore the dimension using ghost particles. When two paths meet, one continues as heir while the other becomes a ghost—continuing independently, invisible to the physical system. The entity count (heirs plus ghosts) remains $n$, enabling a determinant formula.