Strong stationary times for finite Heisenberg walks
Résumé
The random mapping construction of strong stationary times is applied here to finite Heisenberg random walks over $\mathbf{Z}_M$, for odd $M\geq 3$.
When they correspond to $3\times 3$ matrices, the strong stationary times are of order $M^4\ln(M)$, estimate which can be improved to $M^3\ln(M)$
if we are only interested in the convergence to equilibrium of the non-Markovian coordinate in the upper right corner. These results are extended to $N\times N$ matrices, with $N\geq 3$.
All the obtained bounds are believed 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.
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