Hunting a Random Walker on a Low Dimensional Torus

We will define a problem regarding locating a simple random walk on a graph, and give bounds for the optimal solution on toruses of dimension 2,3. 1 Hunting a Random Walker There are several natural variants of the hunting problem, here is one. A bird performs a random walk on a graph. At each step a hunter picks a vertex and either catches the bird or gets to see if the bird visited there before (and possibly when) or not. Will a clever hunter almost surely catch the bird? In particular when the bird performs a fat tail symmetric random walk on Z, how big should the tail be to guarantee escaping with positive probability? Can it escape once transient? One can study versions of the problem for infinite countable Markov chains (or other random processes), in which the question is: will the hunter catch the bird almost surely? Or versions where the hunter is catching the bird almost surely, and the question is how fast the hunter can do it, this can be studied for finite chains as well. Other aspects to consider is the amount of information available for the hunter. Does the hunter get to see the time the bird visited the vertex or just if it ever visited there before, or did the bird visit again since the last time the hunter inspected the vertex. Further limitations on the hunter can involve the size of the memory of the hunter. Benjamini and Kozma (2010) (unpublished), showed that if the underling graph is a d tree and the bird is performing a uniform directed walk away from the root, then there is a hunter that will almost surely catch the bird iff d < 4. In this paper we show an algorithm for the hunter of a SRW on a torus of dimension 2 or 3, and analyze its expected hunting time complexity, comparing it to a simple lower bound on the complexity for any algorithm. The algorithm will need to know a little more than whether the walk visited the vertex, namely it will need to know whether it visited the vertex since the previous examination of this vertex. This is the most limited information in which we found a non trivial solution. The SRW can be replaced by any symmetric walk with steps distribution with a second moment.