Strong stationary times for finite Heisenberg walks

Abstract

The random mapping construction of strong stationary times is applied here to finite Heisenberg random walks over ℤM, for odd M ⩾ 3. When they correspond to 3 × 3 matrices, the strong stationary times are of order M6, estimate which can be improved to M4 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 M2 order. These results are extended to N × 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.