A harmonic function on a graph is a function on its vertices such that the value at every vertex is the average of the values at its neighbors.

Consider the following method for approximating a harmonic function on a graph, given some initial values on each vertex: at each step, pick a random vertex, and replace its value by the average value of its neighbors.

Does this dynamical system have a name? Perhaps something like "Voter Model"? I recall something like "Relaxation" but an internet search gave irrelevant results. Are there texts that deal with this dynamical system? How much is known about it?

I'm interested specifically in the approximation of the value at time $t$ of an arbitrary vertex in the infinite grid $\mathbb{Z}^2$, given some initial conditions. This is a continuous-time random variable, and one can ask whether its expected value at time $t$ is continuous / differentiable / differentiable twice etc.