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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References
Adler, R.L., Weiss, B.: Entropy, a complete metric invariant for automorphisms of the torus. Proc. Natl. Acad. Sci. USA 57, 1573–1576 (1967). https://doi.org/10.1073/pnas.57.6.1573
Asci, C.: Generating uniform random vectors. J. Theor. Probab. 14(2), 333–356 (2001). https://doi.org/10.1023/A:1011155412481
Asci, C.: Generating uniform random vectors in \(Z^{k}_{p}\): the general case. J. Theor. Probab. 22(3), 791–809 (2009). https://doi.org/10.1007/s10959-008-0172-8
Audenaert, K.M.R.: A sharp continuity estimate for the von Neumann entropy. J. Phys. A 40(28), 8127–8136 (2007). https://doi.org/10.1088/1751-8113/40/28/S18
Ben-Hamou, A., Peres, Y.: Cutoff for permuted Markov chains. arXiv preprint arXiv:2104.03568 (2021)
Benoist, Y., Paulin, F.: Systèmes dynamiques élémentaires. Cours de Magistère ENS 3 (2002). https://www.imo.universite-paris-saclay.fr/~paulin/notescours/cours_sysdyn.pdf
Berg, K.R.: On the conjugacy problem for K-systems. PhD thesis, University of Minnesota (1967)
Bordenave, C., Qiu, Y., Zhang, Y.: Spectral gap of sparse bistochastic matrices with exchangeable rows. Ann. Inst. Henri Poincaré Probab. Stat. 56(4), 2971–2995 (2020). https://doi.org/10.1214/20-AIHP1065
Bowen, R.: Markov partitions and minimal sets for Axiom \({\rm A}\) diffeomorphisms. Am. J. Math. 92, 907–918 (1970). https://doi.org/10.2307/2373402
Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Mathematics, revised edn, vol. 470. Springer, Berlin (2008). With a preface by David Ruelle, Edited by Jean-René Chazottes
Chatterjee, S., Diaconis, P.: Speeding up Markov chains with deterministic jumps. Probab. Theory Relat Fields 181(1–3), 377–400 (2021). https://doi.org/10.1007/s00440-021-01049-1
Chung, F.R.K., Diaconis, P., Graham, R.L.: Random walks arising in random number generation. Ann. Probab. 15(3), 1148–1165 (1987)
Diaconis, P.: Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 11. Institute of Mathematical Statistics, Hayward (1988). https://www.jstor.org/stable/4355560
Diaconis, P., Graham, R.: An affine walk on the hypercube. J. Comput. Appl. Math. 41(1–2): 215–235, (1992). https://doi.org/10.1016/0377-0427(92)90251-R
Eberhard, S., Varjú, P.P.: Mixing time of the Chung–Diaconis–Graham random process. Probab. Theory Relat. Fields 179(1–2), 317–344 (2021). https://doi.org/10.1007/s00440-020-01009-1
He, J.: Markov chains on finite fields with deterministic jumps. Electron. J. Probab. 27, 1–17 (2022). https://doi.org/10.1214/22-ejp757
He, J., Pham, H.T., Xu, M.W.: Mixing time of fractional random walk on finite fields. arXiv preprint arXiv:2102.02781 (2021)
Hildebrand, M.: Random processes of the form \(x_{n+1} = a_{n}x_{n} + b_{n} ( ext{mod} p)\). Ann. Probab. 21(2), 710–720 (1993). https://doi.org/10.1214/aop/1176989264
Hildebrand, M.: Random processes of the form \(X_{n+1}=a_{n}X_{n}+b_{n}(\text{ mod } p)\) where \(b_n\) takes on a single value. In: Aldous, D., Pemantle, R. (eds.) Random Discrete Structures (Minneapolis, MN, 1993). The IMA Volumes in Mathematics and Its Applications, vol. 76, pp. 153–174. Springer, New York (1996). https://doi.org/10.1007/978-1-4612-0719-1_10
Hildebrand, M.: On the Chung–Diaconis–Graham random process. Electron. Commun. Probab. 11, 347–356 (2006). https://doi.org/10.1214/ECP.v11-1237
Hildebrand, M.: A lower bound for the Chung–Diaconis–Graham random process. Proc. Am. Math. Soc. 137(4), 1479–1487 (2009). https://doi.org/10.1090/S0002-9939-08-09687-1
Hildebrand, M.: On a lower bound for the Chung–Diaconis–Graham random process. Stat. Probab. Lett. 152, 121–125 (2019). https://doi.org/10.1016/j.spl.2019.04.020
Hildebrand, M.: A multiplicatively symmetrized version of the Chung–Diaconis–Graham random process. J. Theor. Probab. (2021). https://doi.org/10.1007/s10959-021-01088-3
Hildebrand, M., McCollum, J.: Generating random vectors in \(({{\mathbb{Z}}}/p{{\mathbb{Z}}})^{d}\) via an affine random process. J. Theor. Probab. 21(4), 802–811 (2008). https://doi.org/10.1007/s10959-007-0135-5
Klyachko, K.: Random processes of the form \(X_{N}+1= AX_{N}+ B_{N} (\text{ mod } \,p)\). PhD thesis, State University of New York at Albany (2020)
Lang, S.: Algebra. Graduate Texts in Mathematics, vol. 211, 3rd edn. Springer, New York (2002). https://doi.org/10.1007/978-1-4613-0041-0
Lindenstrauss, E., Varjú, P.P.: Spectral gap in the group of affine transformations over prime fields. Ann. Fac. Sci. Toulouse Math. (6) 25(5), 969–993 (2016). https://doi.org/10.5802/afst.1518
Lubotzky, A., Pak, I.: The product replacement algorithm and Kazhdan’s property (T). J. Am. Math. Soc. 14(2), 347–363 (2001). https://doi.org/10.1090/S0894-0347-00-00356-8
Mukherjea, A., Tserpes, N.A.: Measures on Topological Semigroups: Convolution Products and Random Walks. Lecture Notes in Mathematics, vol. 547. Springer, Berlin (1976)
Neville III, R.: On lower bounds of the Chung–Diaconis–Graham random process. PhD thesis, State University of New York at Albany (2011)
Pak, I.: What do we know about the product replacement algorithm? In: Groups and Computation, III (Columbus, OH, 1999). Ohio State University Mathematical Research Institute Publications, vol. 8, pp. 301–347. de Gruyter, Berlin (2001)
Sidorov, N.: Arithmetic dynamics. In: Topics in Dynamics and Ergodic Theory. London Mathematical Society Lecture Note Series, vol. 310, pp. 145–189. Cambridge University Press, Cambridge (2003). https://doi.org/10.1017/CBO9780511546716.010
Tao, T.: Expansion in Finite Simple Groups of Lie Type. Graduate Studies in Mathematics, vol. 164. American Mathematical Society, Providence (2015). https://doi.org/10.1090/gsm/164
Vershik, A.M.: Arithmetic isomorphism of hyperbolic automorphisms of a torus and of sofic shifts. Funktsional. Anal. i Prilozhen. 26(3), 22–27 (1992). https://doi.org/10.1007/BF01075629
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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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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DOI: https://doi.org/10.1007/s00440-022-01128-x
Mathematics Subject Classification
- 60J10
- 37D20