RANDOM WALK ON UNIPOTENT MATRIX GROUPS BY PERSI DIACONIS AND ROBERT HOUGH

We introduce a new method for proving central limit theorems for random walk on nilpotent groups. The method is illustrated in a local central limit theorem on the Heisenberg group, weakening the necessary conditions on the driving measure. As a second illustration, the method is used to study walks on the n×n uni-upper triangular group with entries taken modulo p. The method allows sharp answers to the behavior of individual coordinates: coordinates immediately above the diagonal require order p2 steps for randomness, coordinates on the second diagonal require order p steps; coordinates on the kth diagonal require order p 2 k steps.