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

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

  1. 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.

References

  1. Breuillard, E., Varjú, P.P.: Entropy of Bernoulli convolutions and uniform exponential growth for linear groups. J. Anal. Math. 140(2), 443–481 (2020)

    Article  MathSciNet  Google Scholar 

  2. Breuillard, E., Varjú, P.P.: Irreducibility of random polynomials of large degree. Acta Math. 223(2), 195–249 (2019)

  3. Breuillard, E., Varjú, P. P.: On the Lehmer conjecture and counting in finite fields. Discrete Analysis 2019(5), 8 (2019)

  4. Chung, F.R.K., Diaconis, P., Graham, R.L.: Random walks arising in random number generation. Ann. Probab. 15(3), 1148–1165 (1987)

    Article  MathSciNet  Google Scholar 

  5. Cohen, H.: A course in computational algebraic number theory. Graduate Texts in Mathematics, vol. 138. Springer-Verlag, Berlin (1993)

    Book  Google Scholar 

  6. Diaconis, P.: The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. U.S.A. 93(4), 1659–1664 (1996)

    Article  MathSciNet  Google Scholar 

  7. Dobrowolski, E.: On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith 34(4), 391–401 (1979)

    Article  MathSciNet  Google Scholar 

  8. Erdős, P., Murty, M. R.: On the order of a (mod p). In Number theory (Ottawa, ON, 1996), pp. 87–97 (1999)

  9. Erschler, A.: Isoperimetry for wreath products of Markov chains and multiplicity of selfintersections of random walks. Probab. Theory Related Fields 136(4), 560–586 (2006)

  10. Helfgott, H.A.: Growth and expansion in algebraic groups over finite fields. Analytic methods in arithmetic geometry, 71–111, Contemp. Math., 740, Centre Rech. Math. Proc., Amer. Math. Soc., (2019)

  11. Hildebrand, M.: Random processes of the form \(X_{n+1} = a_{n}X_{n} + b_{n} (\text{ mod } p)\). Ann. Probab. 21(2), 710–720 (1993)

    Article  MathSciNet  Google Scholar 

  12. Hildebrand, M.: On the Chung-Diaconis-Graham random process. Electron. Comm. Probab. 11, 347–356 (2006)

    Article  MathSciNet  Google Scholar 

  13. Hildebrand, M.: On a lower bound for the Chung-Diaconis-Graham random process. Statist. Probab. Lett. 152, 121–125 (2019)

    Article  MathSciNet  Google Scholar 

  14. Hildebrand, M. V.: Rates of convergence of some random processes on finite groups. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-Harvard University (1990)

  15. Konyagin, S.V.: Estimates for Gaussian sums and Waring’s problem modulo a prime. Trudy Mat. Inst. Steklov. 198, 111–124 (1992)

    MATH  Google Scholar 

  16. Lehmer, D.H.: Factorization of certain cyclotomic functions. Ann. of Math. (2) 34(3), 461–479 (1933)

    Article  MathSciNet  Google Scholar 

  17. Lehmer, D.H.: Mathematical methods in large-scale computing units. In Proceedings of a Second Symposium on Large-Scale Digital Calculating Machinery 1949, 141–146 (1951)

  18. Levin, D. A., Peres, Y.: Markov chains and mixing times. American Mathematical Society, Providence, RI. Second edition (2017)

  19. Mahler, K.: An inequality for the discriminant of a polynomial. Michigan Math. J. 11, 257–262 (1964)

    Article  MathSciNet  Google Scholar 

  20. Stark, H.M.: Some effective cases of the Brauer-Siegel theorem. Invent. Math. 23, 135–152 (1974)

    Article  MathSciNet  Google Scholar 

Download references

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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Correspondence to Péter P. Varjú.

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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