Exact probabilities and asymptotics for the one-dimensional coalescing ideal gas

We consider a modification of the well-known system of coalescing random walks in one dimension, both in discrete and continuous time. In our models each particle moves with unit speed, and it can change its direction of movement only at times of collisions with other particles. At these times (and at time 0) the direction is chosen randomly, with equal probability to the left or to the right, independently of anything else. In this article, we exhibit the exact distributions of particle density and of other relevant quantities at finite time t, and their asymptotics as t → ∞. In particular, it appears that the density of particles at time t is equal to the probability of the event that a simple random walk starting at site one first hits the origin after time t It is noteworthy that a relation of the same kind is known to hold for the standard system of coalescing random walks in one dimension, though the proof is quite different in that case.