Consider an oriented graph (e.g. a finite part of the standard grid with some random orientations).

Each minute the following operation takes place: we choose uniformly randomly an ordered pair $(A,B)$ of vertices of the graph, and if there is an oriented path from $A$ to $B$ then we say “success”, take it and invert all the directions of the edges on this path (one can take a random path, or a shortest one). If there is no such a path, we say “failure”.

I guess it might be a classical problem in statistical physics. Anyway, do you have any advice on how to compute the probability of “success” after many steps? Does it converge for any initial directions of the edges? I can imagine two versions of the problem:

(less interesting) compute the probability of the existence of an oriented path between two random vertices on the whole ensemble of edge orientations.

(more interesting) We perform the above operation many many times, what is the frequency of success?