Strong Stationary Times for Finite Heisenberg Walks

The random mapping construction of strong stationary times is applied here to finite Heisenberg random walks over DOUBLE-STRUCK CAPITAL Z(M), for odd M > 3. When they correspond to 3 x 3 matrices, the strong stationary times are of order M-6, estimate which can be improved to M-4 if we are only interested in the convergence to equilibrium of the last column. Simulations by Chhaibi suggest that the proposed strong stationary time is of the right M-2 order. These results are extended to N x N matrices, with N > 3. All the obtained bounds are thought to be non-optimal, nevertheless this original approach is promising, as it relates the investigation of the previously elusive strong stationary times of such random walks to new absorbing Markov chains with a statistical physics flavor and whose quantitative study is to be pushed further. In addition, for N = 3, a strong equilibrium time is proposed in the same spirit for the non-Markovian coordinate in the upper right corner. This result would extend to separation discrepancy the corresponding fast convergence for this coordinate in total variation and open a new method for the investigation of this phenomenon in higher dimension.