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Cut-off phenomenon for the ax+b Markov chain over a finite field

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Emmanuel Breuillard
Emmanuel Breuillard
Emmanuel Breuillard
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Péter P. Varjú
Péter P. Varjú
Péter P. Varjú
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Abstract

We study the Markov chain xn+1=axn+bnx_{n+1}=ax_n+b_n on a finite field Fp{\mathbb {F}}_p, where a∈Fp×a \in {\mathbb {F}}_p^{\times } is fixed and bnb_n are independent and identically distributed random variables in Fp{\mathbb {F}}_p. Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show that the chain exhibits a cut-off phenomenon for most primes p and most values of a∈Fp×a \in {\mathbb {F}}_p^\times . We also obtain weaker, but unconditional, upper bounds for the mixing time.
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Probability Theory and Related Fields (2022) 184:85–113
https://doi.org/10.1007/s00440-022-01161-w
Cut-off phenomenon for the ax+b Markov chain over a
finite field
Emmanuel Breuillard1·Péter P. Varjú2
Received: 17 October 2019 / Revised: 17 March 2022 / Accepted: 13 August 2022 /
Published online: 2 September 2022
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
Abstract
We study the Markov chain xn+1=axn+bnon a finite field Fp, where a∈F×
p
is fixed and bnare independent and identically distributed random variables in Fp.
Conditionally on the Riemann hypothesis for all Dedekind zeta functions, we show
that the chain exhibits a cut-off phenomenon for most primes pand most values of
a∈F×
p. We also obtain weaker, but unconditional, upper bounds for the mixing time.
Mathematics Subject Classification Primary 60J10 ·Secondary 11T23
1 Introduction
Let pbe a prime number and Fpthe field with pelements. Consider the Markov chain
on Fp
xn+1=axn+bn,(1.1)
where the multiplier ais non-zero and the bn’s are independent random variables
taking values in Fpwith a common law μ. We will assume that the support of μhas at
EB has received funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No. 617129); PV has received
funding from the Royal Society and the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation programme (grant agreement No. 803711).
BPéter P. Varjú
pv270@dpmms.cam.ac.uk
Emmanuel Breuillard
breuillard@maths.ox.ac.uk
1Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
2Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3
0WB, UK
123
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... Our strongest results require various assumptions on the parameters, but we are able to obtain results requiring only q = p a prime with p ≤ exp(n 1/13 ) where n is the degree of the polynomial. Our proofs use Fourier analysis, and rely on tools recently applied by Breuillard and Varjú [5,6] to study the ax+b process, which show equidistribution for f (α) at a single point. We extend this to handle multiple roots and the Hasse derivatives of f , which allow us to study the irreducible factors with multiplicity. ...
... Note that the value of f (α) for some α ∈ F q d can be viewed as the state of a random walk defined by X t+1 = αX t + ε t+1 where ε t+1 is drawn from some non trivial distribution. This is known as the ax + b process or the Chung-Diaconis-Graham process, and we use recent tools developed to study this process in [5,6]. For this reason, our strongest results apply only when the coefficients lie in F p . ...
... To study the N i (f ), which count irreducible factors with multiplicity, we instead study the equidistribution of the values of f along with its Hasse derivatives. This requires extending the tools developed in [5,6]. ...
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