Lattice random walks for sets of random walkers. First passage times

We have studied the mean first passage time for the first of aset of random walkers to reach a given lattice point on infinite lattices ofD dimensions. In contrast to the well-known result ofinfinite mean first passage times for one random walker in all dimensionsD, we findfinite mean first passage times for certain well-specified sets of random walkers in all dimensions, exceptD = 2. The number of walkers required to achieve a finite mean time for the first walker to reach the given lattice point is a function of the lattice dimensionD. ForD > 4, we find that only one random walker is required to yield a finite first passage time, provided that this random walker reaches the given lattice point with unit probability. We have thus found a simple random walk property which sticks atD > 4.Supported in part by a grant from Charles and Renée Taubman and by the National Science Foundation, Grant CHE78-21460.