RANDOM WALKS ON GROUPS: STRONG UNIFORM TIME APPROACH

mixing. We show that the total separation is the mean of the best possible strong uniform time. We prove various bounds on the total separation, find connections with hitting times and establish relations between total separations under several natural operations on walks on groups, such as rescaling of the walk, taking direct and wreath product of groups. In this work the emphasis is given to the study of concrete examples of walks. The successful applications of the method include finding sharp bounds on the total separation for the natural random walks on cube, cyclic group, dihedral group, sym- metric group, hyperoctahedral group, Heisenberg group, and others. In several cases we were able to obtain not only sharp bounds, but find the exact value of the total separation.