• Bernoulli
  • Volume 24, Number 2 (2018), 993-1009.

Mixing time and cutoff for a random walk on the ring of integers mod $n$

Michael Bate and Stephen Connor

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We analyse a random walk on the ring of integers mod $n$, which at each time point can make an additive ‘step’ or a multiplicative ‘jump’. When the probability of making a jump tends to zero as an appropriate power of $n$, we prove the existence of a total variation pre-cutoff for this walk. In addition, we show that the process obtained by subsampling our walk at jump times exhibits a true cutoff, with mixing time dependent on whether the step distribution has zero mean.

Article information

Bernoulli, Volume 24, Number 2 (2018), 993-1009.

Received: December 2015
Revised: February 2016
First available in Project Euclid: 21 September 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

cutoff phenomenon group representation theory mixing time pre-cutoff random number generation random walk


Bate, Michael; Connor, Stephen. Mixing time and cutoff for a random walk on the ring of integers mod $n$. Bernoulli 24 (2018), no. 2, 993--1009. doi:10.3150/16-BEJ832.

Export citation


  • [1] Apostol, T.M. (1974). Mathematical Analysis, 2nd ed. Reading, MA: Addison-Wesley.
  • [2] Chung, F.R.K., Diaconis, P. and Graham, R.L. (1987). Random walks arising in random number generation. Ann. Probab. 15 1148–1165.
  • [3] Connor, S.B. (2010). Separation and coupling cutoffs for tuples of independent Markov processes. ALEA Lat. Am. J. Probab. Math. Stat. 7 65–77.
  • [4] Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes – Monograph Series 11. Hayward, CA: IMS.
  • [5] Diaconis, P. (2011). The mathematics of mixing things up. J. Stat. Phys. 144 445–458.
  • [6] Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159–179.
  • [7] Hildebrand, M. (1990). Rates of convergence of some random processes on finite groups. Ph.D. thesis, Dept. Mathematics, Harvard University.
  • [8] Hildebrand, M. (1993). Random processes of the form $X_{n+1}=a_{n}X_{n}+b_{n}\pmod{p}$. Ann. Probab. 21 710–720.
  • [9] Hildebrand, M. (1994). Random walks supported on random points of $\mathbf{Z}/n\mathbf{Z}$. Probab. Theory Related Fields 100 191–203.
  • [10] Hildebrand, M. (1994). Some random processes related to affine random walks. IMA preprint 1210.
  • [11] Levin, D.A., Peres, Y. and Wilmer, E.L. (2009). Markov Chains and Mixing Times. Providence, RI: Amer. Math. Soc. With a chapter by James G. Propp and David B. Wilson.
  • [12] Mossel, E., Peres, Y. and Sinclair, A. (2004). Shuffling by semi-random transpositions. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS’04) 572–581. Washington, DC, USA: IEEE Computer Society.
  • [13] Saloff-Coste, L. (2004). Random walks on finite groups. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 263–346. Berlin: Springer.
  • [14] Smith, A. (2014). A Gibbs sampler on the $n$-simplex. Ann. Appl. Probab. 24 114–130.
  • [15] Subag, E. (2013). A lower bound for the mixing time of the random-to-random insertions shuffle. Electron. J. Probab. 18 1–20.
  • [16] Terras, A. (1999). Fourier Analysis on Finite Groups and Applications. London Mathematical Society Student Texts 43. Cambridge: Cambridge Univ. Press.