Rendezvous in higher dimensions

Two players are placed on the integer lattice Zn (consisting of points in n-dimensional space with all coordinates integers) so that their vector difference is of length 2 and parallel to one of the axes. Their aim is to move to an adjacent node in each period, so that they meet (occupy same node) in least expected time R(n), called the rendezvous value. We assume they have no common notion of directions or orientations (i.e., no common notion of "clockwise"). We extend the known result R(1)=3.25 of Alpern and Gal to obtain R(2)=197/32=\allowbreak6.\allowbreak16, and the bounds 2n≤R(n)≤(32n3+12n2−2n−3)/12n2. For n=2 we characterize the set of all optimal strategies and show that none of them simultaneously maximizes the probability of meeting by time t for all t. This behavior differs from that found by Anderson and Fekete, and the authors, for the related problem where the players are initially placed at diagonals of one of the squares of the lattice Z2.