Abstract
We study the Markov chain \(x_{n+1}=ax_n+b_n\) on a finite field \({\mathbb {F}}_p\), where \(a \in {\mathbb {F}}_p^{\times }\) is fixed and \(b_n\) are independent and identically distributed random variables in \({\mathbb {F}}_p\). 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 \(a \in {\mathbb {F}}_p^\times \). We also obtain weaker, but unconditional, upper bounds for the mixing time.
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Notes
Here and everywhere in this paper, when we say that we assume GRH, we mean that we assume that the Dedekind zeta function of an arbitrary number field satisfies the Riemann hypothesis, namely has its non trivial zeroes on the critical line.
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Acknowledgements
It is a pleasure to thank Boris Bukh, Persi Diaconis, Jonathan Hermon and Ariel Rapaport for helpful conversations. The first named author is grateful to the Newton Institute in Cambridge for its hospitality while part of this work was conducted. We would also like to thank the referee for their very careful examination of our manuscript.
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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)
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Breuillard, E., Varjú, P.P. Cut-off phenomenon for the ax+b Markov chain over a finite field. Probab. Theory Relat. Fields 184, 85–113 (2022). https://doi.org/10.1007/s00440-022-01161-w
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DOI: https://doi.org/10.1007/s00440-022-01161-w
Mathematics Subject Classification
- Primary 60J10
- Secondary 11T23