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Article Dans Une Revue Probability Theory and Related Fields Année : 2022

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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Dates et versions

hal-03016958 , version 1 (20-11-2020)
hal-03016958 , version 2 (04-07-2021)
hal-03016958 , version 3 (16-10-2023)

Identifiants

Citer

Laurent Miclo. Strong stationary times for finite Heisenberg walks. Probability Theory and Related Fields, 2022, 1-2, pp.275-322. ⟨10.1007/s00440-022-01147-8⟩. ⟨hal-03016958v3⟩
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