Description

In the voter experiment we start with an \(n \times n\) rectangular lattice of sites (referred to as voters): \[ V = \{0, 1, \ldots, n - 1\}^2 \] Each element of \(V\) has four neighbors; the neighbors of \((i, j)\) are \(\{(i - 1, j), (i + 1, j), (i, j - 1), (i, j + 1)\}\) where the arithmetic operations are interpreted modulo \(n\):

• \((n - 1) + 1 = 0 \mod n\)
• \(0 - 1 = n - 1 \mod n\)

With this neighborhood structure, our set of sites is topologically a torus, a doughnut-shaped surface Each site can be in any of \(k\) different states, denoted by colors. Initially, the each site is randomly given one of the colors. Time is discrete, and the dynamics of the voter process are as follows: at each time \(t \in \N\),

1. A site is chosen at random (each site is equally likely to be chosen),
2. A neighbor of this site is chosen at random.
3. The state of the selected site is changed to the state of the selected neighbor.

The random variables of interest are the counts for each state (color), along with the total number of colors remaining. The simulation stops automatically when a single color remains. An additional stop option allows the simulation to continue until this happens. The parameters \(n\) and \(k\) can be varied with the scrollbars.