Strong stationary times for finite Heisenberg walk

The random mapping construction of strong stationary times is applied here to finite Heisenberg random walks over $\ZZ_M$, for odd $M\geq 3$. When they correspond to $3\times 3$ 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 Chhaïbi suggest that the proposed strong stationary time is of the right $M^2$ order. These results are extended to $N\times N$ matrices, with $N\geq 3$. 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.