Strong stationary times for finite Heisenberg walks

¹ Toulouse School of Economics 1, Esplanade de l’Université 31080 Toulouse Cedex 06, France
² Institut de Mathématiques de Toulouse Université Paul Sabatier, 118, route de Narbonne 31062 Toulouse cedex 9, France

^* Corresponding author: miclo@math.cnrs.fr

Received: 4 July 2021
Accepted: 27 March 2023

Abstract

The random mapping construction of strong stationary times is applied here to finite Heisenberg random walks over ℤ_M, 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.