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Random walks on the BMW monoid: an algebraic approach

  • Sarah WolffEmail author
Article
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Abstract

We consider Metropolis-based systematic scan algorithms for generating Birman–Murakami–Wenzl (BMW) monoid basis elements of the BMW algebra. As the BMW monoid consists of tangle diagrams, these scanning strategies can be rephrased as random walks on links and tangles. We also consider the Brauer algebra and use Metropolis-based scans to generate Brauer diagrams, giving rise to random walks on perfect matchings. Taking an algebraic perspective, we translate these walks into left multiplication operators in the BMW algebra and so give an algebraic interpretation of the Metropolis algorithm in this setting.

Keywords

Metropolis algorithm Systematic scans Random walks Representation theory Semisimple algebras 

Notes

Acknowledgements

The author would like to especially thank Arun Ram for his interest in the work and many helpful conversations, as well as Daniel Rockmore for his guidance and support. The author would also like to thank two anonymous referees for the many helpful suggestions that greatly improved the paper.

References

  1. 1.
    Aldous, D., Fill, J.A.: Reversible markov chains and random walks on graphs. Unfinished monograph, recompiled 2014 (2002).http://www.stat.berkeley.edu/~aldous/RWG/book.html
  2. 2.
    Benkart, G., Ram, A., Shader, C.L.: Tensor product representations for orthosymplectic Lie superalgebras. J. Pure Appl. Algebra 130(1), 1–48 (1998)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bergeron, F., Favreau, L.: Fourier transform over semi-simple algebras and harmonic analysis for probabilistic algorithms. Discrete Math., 139(1-3):19–32. Formal power series and algebraic combinatorics (Montreal, PQ, 1992) (1995)Google Scholar
  4. 4.
    Brémaud, P.: Markov chains, volume 31 of texts in applied mathematics. In: Marsden, J.E., Sirovich, L., Golubitsky, M., Jäger, W. (eds.) Gibbs fields, Monte Carlo simulation, and queues. Springer, New York (1999)zbMATHGoogle Scholar
  5. 5.
    Cai, Y.: How rates of convergence for Gibbs fields depend on the interaction and the kind of scanning used. In: Hou, Z., Filar, J.A., Chen, A. (eds.) Markov Processes and Controlled Markov Chains (Changsha, 1999), pp 489–498. Kluwer Acad. Publ, Dordrecht (2002)Google Scholar
  6. 6.
    Ceccherini-Silberstein, T., Scarabotti, F., Tolli, F.: Harmonic Analysis on Finite Groups, Volume 108 of Cambridge Studies in Advanced Mathematics. Representation Theory, Gelfand Pairs and Markov Chains. Cambridge University Press, Cambridge (2008)zbMATHGoogle Scholar
  7. 7.
    Clausen, M.: Fast generalized Fourier transforms. Theoret. Comput. Sci. 67(1), 55–63 (1989)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Clausen, M., Baum, U.: Fast Fourier Transforms. Bibliographisches Institut, Mannheim (1993)zbMATHGoogle Scholar
  9. 9.
    Diaconis, P.: Group representations in probability and statistics. Institute of Mathematical Statistics Lecture Notes-Monograph Series, 11. Institute of Mathematical Statistics., Hayward, CA (1988)Google Scholar
  10. 10.
    Diaconis, P., Ram, A.: Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques. Michigan Math. J. 48, 157–190 (2000)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Diaconis, P., Saloff-Coste, L.: Random walks on finite groups: a survey of analytic techniques. In: Heyer, H. (ed.) Probability Measures on Groups and Related Structures. XI (Oberwolfach, 1994), pp. 44–75. World Sci. Publ, River Edge (1995)Google Scholar
  12. 12.
    Fishman, G.: Coordinate selection rules for Gibbs sampling. Ann. Appl. Probab. 6(2), 444–465 (1996)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Goodman, F., Hauschild, H.: Affine Birman-Wenzl-Murakami algebras and tangles in the solid torus. Fundam. Math. 190, 77–137 (2006)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hammersley, J.M., Handscomb, D.C.: Monte Carlo Methods. Methuen & Co., Ltd., Barnes & Noble, Inc., London, New York (1965)zbMATHGoogle Scholar
  15. 15.
    Karlin, S.: A First Course in Stochastic Processes. Academic Press, New York (1966)Google Scholar
  16. 16.
    Liu, J.: Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. Springer, New York (2008)zbMATHGoogle Scholar
  17. 17.
    Maslen, D., Rockmore, D.: Separation of variables and the computation of Fourier transforms on finite groups. I. J. Amer. Math. Soc. 10(1), 169–214 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Maslen, D., Rockmore, D.N., Wolff, S.: The efficient computation of fourier transforms on semisimple algebras. J. Fourier Anal. Appl. 24, 1377–1400 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Metropolis, N., Rosenbluth, A., Rosenbluth, M., Teller, A., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21(1087), 1087–1092 (1953)CrossRefGoogle Scholar
  20. 20.
    Morton, H., Wasserman, A.: A basis for the Birman-Wenzl algebra. Revised 2000. Unpublished Manuscript (1989)Google Scholar
  21. 21.
    Nazarov, M.: Young’s orthogonal form for Brauer’s centralizer algebra. J. Algebra 182(3), 664–693 (1996)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Saloff-Coste, L.: Random walks on finite groups. In: Kesten, H. (ed.) Probability on Discrete Structures, Volume 110 of Encyclopaedia Math. Sci., pp. 263–346. Springer, Berlin (2004)CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Denison UniversityGranvilleUSA

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