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Accelerating abelian random walks with hyperbolic dynamics

Abstract

Given integers \(d \ge 2, n \ge 1\), we consider affine random walks on torii \((\mathbb {Z}/ n \mathbb {Z})^{d}\) defined as \(X_{t+1} = A X_{t} + B_{t} \mod n\), where \(A \in \mathrm {GL}_{d}(\mathbb {Z})\) is a invertible matrix with integer entries and \((B_{t})_{t \ge 0}\) is a sequence of iid random increments on \(\mathbb {Z}^{d}\). We show that when A has no eigenvalues of modulus 1, this random walk mixes in \(O(\log n \log \log n)\) steps as \(n \rightarrow \infty \), and mixes actually in \(O(\log n)\) steps only for almost all n. These results are similar to those of Chung et al. (Ann Probab 15(3):1148–1165, 1987) on the so-called Chung–Diaconis–Graham process, which corresponds to the case \(d=1\). Our proof is based on the initial arguments of Chung, Diaconis and Graham, and relies extensively on the properties of the dynamical system \(x \mapsto A^{\top } x\) on the continuous torus \(\mathbb {R}^{d} / \mathbb {Z}^{d}\). Having no eigenvalue of modulus one makes this dynamical system a hyperbolic toral automorphism, a typical example of a chaotic system known to have a rich behaviour. As such our proof sheds new light on the speed-up gained by applying a deterministic map to a Markov chain.

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Acknowledgements

We thank Ioannis Iakovoglou for useful discussions. We also thank the anonymous referees for pointing out a gap in a proof in the first version of this manuscript and for reference [25].

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Correspondence to Bastien Dubail.

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Dubail, B., Massoulié, L. Accelerating abelian random walks with hyperbolic dynamics. Probab. Theory Relat. Fields 184, 939–968 (2022). https://doi.org/10.1007/s00440-022-01128-x

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Mathematics Subject Classification

  • 60J10
  • 37D20