The less obvious question is: What is the diffusive scaling limit of the joint collection of one-dimensional coalescing random walks starting from every point in space-time?
Formally speaking, it is a collection of one-dimensional coalescing Brownian motions starting from every space-time point in
The main difficulty in proving convergence stems from the existence of random points from which the limiting object can have multiple paths.
Arratia and Tóth and Werner were aware of the existence of such points and they provided different conventions to avoid such multiplicity.
The introduction of this topology allowed them to prove the convergence of the coalescing random walks to a unique limiting object and characterize it.
Graphical construction of the voter model with configuration
. The arrows determine when a voter changes its opinion to that of the neighbor pointed to by the arrow. The genealogies are obtained by following the arrows backwards in time, which are distributed as coalescing random walks.
Coalescing random walks on the discrete space-time lattice
From each lattice point, an arrow is drawn either up-right or up-left with probability 1/2 each. The random walks move upward in time by following the arrows, and different random walks coalesce once they meet.
