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Random Motion on Finite Rings, I: Commutative Rings

  • Arvind AyyerEmail author
  • Pooja Singla
Article
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Abstract

We consider irreversible Markov chains on finite commutative rings randomly generated using both addition and multiplication. We restrict ourselves to the case where the addition is uniformly random and multiplication is arbitrary. We first prove formulas for eigenvalues and multiplicities of the transition matrices of these chains using the character theory of finite abelian groups. The examples of principal ideal rings (such as \(\mathbb {Z}_{n})\) and finite chain rings (such as \(\mathbb {Z}_{p^{k}})\) are particularly illuminating and are treated separately. We then prove a recursive formula for the stationary probabilities for any ring, and use it to prove explicit formulas for the probabilities for finite chain rings when multiplication is also uniformly random. Finally, we prove constant mixing time for our chains using coupling.

Keywords

Finite commutative rings Markov chains Semigroup algebras Spectrum Stationary distribution Mixing time Finite chain rings 

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Notes

Acknowledgements

We are very grateful to the anonymous referees for many constructive suggestions. We would also like to thank M. Krishnapur and B. Steinberg for enlightening discussions. The authors would like to acknowledge support in part by a UGC Centre for Advanced Study grant. The first author (AA) would like to acknowledge support from Department of Science and Technology grants DST/INT/SWD/VR/P-01/2014 and EMR/2016/006624.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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