Two weeks ago we asked:
Ruth and Ron start together at the origin and take a walk on the integers. Every day they make a move. They take turns in flipping a coin and they move together right or left according to the outcome. Their coin flips create a simple random walk starting at the origin on the integers.
We know for sure that they we will return to the origin infinitely many times. However, their random walk never comes back to the origin, so we know for sure that one of them did not follow the rules!
Test your intuition: Is it possible to figure out from the walk whether it was Ruth or Ron who did not follow the coin-flipping rule?
And the answer is yes! This is proved in the following paper:
Identifying the Deviator, by Noga Alon, Benjamin Gunby, Xiaoyu He, Eran Shmaya, and Eilon Solan
Abstract: A group of players are supposed to follow a prescribed profile of strategies. If they follow this profile, they will reach a given target. We show that if the target is not reached because some player deviates, then an outside observer can identify the deviator. We also construct identification methods in two nontrivial cases.
The paper gives a non-constructive proof for a strategy to identify the deviator and in the case of two-player random walk it gives an ingenious explicit identification method!
Here is a variant of the problem devoted to Itai Benjamini: Consider planar percolation on the rectangular grid. One player chooses every vertical edge with probability 1/2 and one player chooses every horizontal edge with probability 1/2. Lo and behold, there is an infinite open cluster. Can you (explicitely) detect the deviator?