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Mixing time of the Chung–Diaconis–Graham random process

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Sean K. Eberhard at University of Warwick
Sean K. Eberhard
Péter P. Varjú
Péter P. Varjú
Péter P. Varjú
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Abstract

Define (Xn) on Z/qZ by Xn+1=2Xn+bn, where the steps bn are chosen independently at random from -1,0,+1. The mixing time of this random walk is known to be at most 1.02log2q for almost all odd q (Chung, Diaconis, Graham in Ann Probab 15(3):1148–1165, 1987), and at least 1.004log2q (Hildebrand in Proc Am Math Soc 137(4):1479–1487, 2009). We identify a constant c=1.01136⋯ such that the mixing time is (c+o(1))log2q for almost all odd q. In general, the mixing time of the Markov chain Xn+1=aXn+bn modulo q, where a is a fixed positive integer and the steps bn are i.i.d. with some given distribution in Z, is related to the entropy of a corresponding self-similar Cantor-like measure (such as a Bernoulli convolution). We estimate the mixing time up to a 1+o(1) factor whenever the entropy exceeds (loga)/2.
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Probability Theory and Related Fields (2021) 179:317–344
https://doi.org/10.1007/s00440-020-01009-1
Mixing time of the Chung–Diaconis–Graham random
process
Sean Eberhard1·Péter P. Varjú1
Received: 19 March 2020 / Revised: 21 September 2020 / Accepted: 28 September 2020 /
Published online: 10 October 2020
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract
Define (Xn)on Z/qZby Xn+1=2Xn+bn, where the steps bnare chosen indepen-
dently at random from −1,0,+1. The mixing time of this random walk is known to
be at most 1.02 log2qfor almost all odd q(Chung, Diaconis, Graham in Ann Probab
15(3):1148–1165, 1987), and at least 1.004 log2q(Hildebrand in Proc Am Math Soc
137(4):1479–1487, 2009). We identify a constant c=1.01136 ... such that the mix-
ing time is (c+o(1)) log2qfor almost all odd q. In general, the mixing time of the
Markov chain Xn+1=aXn+bnmodulo q, where ais a fixed positive integer and
the steps bnare i.i.d. with some given distribution in Z, is related to the entropy of
a corresponding self-similar Cantor-like measure (such as a Bernoulli convolution).
We estimate the mixing time up to a 1 +o(1)factor whenever the entropy exceeds
(log a)/2.
Mathematics Subject Classification Primary 60J10; Secondary 11N36
1 Introduction
Let a>1 be a positive integer and let μbe a finitely supported measure on Z. Assume
that gcd(supp μ−supp μ) =1, i.e., that μis not supported on a coset of a proper
subgroup. Define a random process (Xn)n≥0on Zby X0=0 and
Xn+1=aXn+bn,
Sean Eberhard and Péter P. Varjú have received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No.
803711). PV was supported by the Royal Society.
BPéter P. Varjú
pv270@dpmms.cam.ac.uk
Sean Eberhard
eberhard@maths.cam.ac.uk
1Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
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