We study a random walk where all steps have same length but different directions. Using recent results from inverse Littlewood-Offord theory (of Nguyen, Tao, and Vu) we obtain a sharp bound for the maximum returning probability of such random walks, for any dimension other than $d=3$. In the euclidean space, we pose a new conjecture in Discrete Geometry, from which an analogous bound would follow.
Joint work with Van Vu.