Electronic Journal of Probability

Mixing and cut-off in cycle walks

Robert Hough

Full-text: Open access

Abstract

Given a sequence $(\mathfrak{X} _i, \mathscr{K} _i)_{i=1}^\infty $ of Markov chains, the cut-off phenomenon describes a period of transition to stationarity which is asymptotically lower order than the mixing time. We study mixing times and the cut-off phenomenon in the total variation metric in the case of random walk on the groups $\mathbb{Z} /p\mathbb{Z} $, $p$ prime, with driving measure uniform on a symmetric generating set $A \subset \mathbb{Z} /p\mathbb{Z} $.

Article information

Source
Electron. J. Probab., Volume 22 (2017), paper no. 90, 49 pp.

Dates
Received: 20 June 2016
Accepted: 15 September 2017
First available in Project Euclid: 18 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1508292261

Digital Object Identifier
doi:10.1214/17-EJP108

Mathematical Reviews number (MathSciNet)
MR3718718

Zentralblatt MATH identifier
1378.60073

Subjects
Primary: 60B10: Convergence of probability measures
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60G50: Sums of independent random variables; random walks 60J60: Diffusion processes [See also 58J65] 11H06: Lattices and convex bodies [See also 11P21, 52C05, 52C07] 11A63: Radix representation; digital problems {For metric results, see 11K16}

Keywords
random walk on a group random lattice cut-off phenomenon embedded hypercube

Rights
Creative Commons Attribution 4.0 International License.

Citation

Hough, Robert. Mixing and cut-off in cycle walks. Electron. J. Probab. 22 (2017), paper no. 90, 49 pp. doi:10.1214/17-EJP108. https://projecteuclid.org/euclid.ejp/1508292261


Export citation

References

  • [1] David Aldous and Persi Diaconis, Shuffling cards and stopping times, Amer. Math. Monthly 93 (1986), no. 5, 333–348.
  • [2] Michael Bate and Stephen Connor, Cutoff for a random walk on the integers mod $n$, 2014.
  • [3] Henry Cohn and Yufei Zhao, Sphere packing bounds via spherical codes, Duke Math. J. 163 (2014), no. 10, 1965–2002.
  • [4] Amir Dembo, Jian Ding, Jason Miller, and Yuval Peres, Cut-off for lamplighter chains on tori: dimension interpolation and phase transition, 2013.
  • [5] P. Diaconis and L. Saloff-Coste, Moderate growth and random walk on finite groups, Geom. Funct. Anal. 4 (1994), no. 1, 1–36.
  • [6] Persi Diaconis, Group representations in probability and statistics, Lecture Notes-Monograph Series 11 (1988), i–192.
  • [7] Persi Diaconis, Threads through group theory, Character theory of finite groups 524 (2010), 33–47.
  • [8] Persi Diaconis, R. L. Graham, and J. A. Morrison, Asymptotic analysis of a random walk on a hypercube with many dimensions, Random Structures Algorithms 1 (1990), no. 1, 51–72.
  • [9] Persi Diaconis and Robert Hough, Random walk on unipotent matrix groups, http://arXiv.org/abs/1512.06304arXiv:1512.06304 (2015).
  • [10] Persi Diaconis and Laurent Saloff-Coste, Separation cut-offs for birth and death chains, Ann. Appl. Probab. 16 (2006), no. 4, 2098–2122.
  • [11] Persi Diaconis and Mehrdad Shahshahani, Time to reach stationarity in the Bernoulli–Laplace diffusion model, SIAM Journal on Mathematical Analysis 18 (1987), no. 1, 208–218.
  • [12] Jian Ding, Eyal Lubetzky, and Yuval Peres, Total variation cutoff in birth-and-death chains, Probab. Theory Related Fields 146 (2010), no. 1-2, 61–85.
  • [13] Carl Dou and Martin Hildebrand, Enumeration and random random walks on finite groups, Ann. Probab. 24 (1996), no. 2, 987–1000.
  • [14] Ben Green and Terence Tao, The quantitative behaviour of polynomial orbits on nilmanifolds, Annals of Mathematics 175 (2012), no. 2, 465–540.
  • [15] Andrew Simon Greenhalgh, Random walks on groups with subgroup invariance properties, ProQuest LLC, Ann Arbor, MI, 1989, Thesis (Ph.D.)–Stanford University.
  • [16] Martin Hildebrand, Random walks supported on random points of $\mathbf Z/n\mathbf Z$, Probab. Theory Related Fields 100 (1994), no. 2, 191–203.
  • [17] Daniel Jerison, Lionel Levine, and John Pike, Mixing time and eigenvalues of the abelian sandpile Markov chain, http://arXiv.org/abs/1511.00666arXiv:1511.00666 (2015).
  • [18] G. A. Kabatjanskiĭ and V. I. Levenšteĭn, Bounds for packings on the sphere and in space, Problemy Peredači Informacii 14 (1978), no. 1, 3–25.
  • [19] Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Classics in Mathematics, Springer-Verlag, Berlin, 2011, Isoperimetry and processes, Reprint of the 1991 edition.
  • [20] DA Levin, Yuval Peres, and Elizabeth Wilmer, Markov chains and mixing times, American Mathematical Society, 2009.
  • [21] Eyal Lubetzky and Yuval Peres, Cutoff on all Ramanujan graphs, http://arXiv.org/abs/1507.04725arXiv:1507.04725 (2015).
  • [22] C. A. Rogers, Mean values over the space of lattices, Acta Math. 94 (1955), 249–287.
  • [23] Carl Ludwig Siegel, A mean value theorem in geometry of numbers, Ann. of Math. (2) 46 (1945), 340–347.
  • [24] Carl Ludwig Siegel, Lectures on the geometry of numbers, Springer Science & Business Media, 2013.
  • [25] Anders Södergren, On the distribution of angles between the N shortest vectors in a random lattice, Journal of the London Mathematical Society (2011), jdr032.
  • [26] Andrey Sokolov, Andrew Melatos, Tien Kieu, and Rachel Webster, Memory on multiple time-scales in an abelian sandpile, Physica A: Statistical Mechanics and its Applications 428 (2015), 295–301.
  • [27] Bálint Virág, Random walks on finite convex sets of lattice points, J. Theoret. Probab. 11 (1998), no. 4, 935–951.
  • [28] David Bruce Wilson, Random random walks on ${\mathbf Z}^{d}_{2}$, Probab. Theory Related Fields 108 (1997), no. 4, 441–457.