Excited Random Walk in Three Dimensions Has Positive Speed

Excited random walk is a model of a random walk on Z which, whenever it encounters a new vertex it receives a push toward a specific direction, call it the “right”, while when it reaches a vertex it “already knows”, it performs a simple random walk. This model has been suggested in [BW] and had since got lots of attention, see [V, Z]. The reason for the interest is that it is situated very naturally between two classical models: random walk in random environment and reinforced random walk. A reinforced random walk is a walk on a graph (say Z) that, whenever it passes through an edge, it changes the weight of this edge, usually positively (i.e. the edge has now a greater probability to be chosen when the random walk rereaches one of its end points) but possibly also negatively, with the extreme being the “bridge-burning random walk” that can never traverse the same edge twice. The problem appears naturally in brain research in connection with the evolution of neural networks. Reinforced random walk models are notoriously difficult to analyze, and even the question whether the simplest one-reinforced random walk on Z is recurrent or transient is open. See [K90, KR99, PV99, DKL02] for some known results. Random walk in a random environment is also a model in which the environment is random, but independently of the walk. For example, one may throw a coin at every point of Z to decide if at this point thewalk will have a push to the left or to the right, and then perform random walk on the resulting weighted graph. The independence of the walk from the environment turns out to be a powerful leverage, and many very precise results are known. See e.g. the book [H95]. Excited random walk has, seemingly, all the difficulties of reinforced random walk: the environment depends on the walk, and in a dynamic way. However, it has two significant advantages. The first is the inherent directedness: the drift of excited random walk is always in the same direction, and in particular, it can be coupled with simple random walk so that the excited is always to the right of the simple random walk. The second is the projected simple random walk of lower dimension. Thus, for example, for the excited random walk in three dimensions, its projection on the two directions orthogonal to our “right” is a simple two-dimensional random walk, up to a time change. Thus, for example, it is clear that the three dimensional excited random walk is transient. Indeed, since a two-dimensional simple random walk visits an order of n/ logn vertices, the three dimensional excited random walk must visit at least n/ logn vertices. This means, roughly, that R(n)1 > n/ logn − C √ n log logn (x1 denoting the first, “left-right” coordinate of x), and in particular that R(n) drifts to the right and returns to every point only a finite number of times (in the two