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Mixing of the upper triangular matrix walk

Abstract

We study a natural random walk over the upper triangular matrices, with entries in the field \({\mathbb{Z}_2}\) , generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n 2) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields \({\mathbb{Z}_q}\) for q prime.

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Correspondence to Allan Sly.

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Peres, Y., Sly, A. Mixing of the upper triangular matrix walk. Probab. Theory Relat. Fields 156, 581–591 (2013). https://doi.org/10.1007/s00440-012-0436-1

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Keywords

  • Mixing time
  • Random walks on groups

Mathematics Subject Classification

  • 60J10