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Commutation relations and Markov chains
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  • Published: 13 February 2008

Commutation relations and Markov chains

  • Jason Fulman1 

Probability Theory and Related Fields volume 144, pages 99–136 (2009)Cite this article

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Abstract

It is shown that the combinatorics of commutation relations is well suited for analyzing the convergence rate of certain Markov chains. Examples studied include random walk on irreducible representations, a local random walk on partitions whose stationary distribution is the Ewens distribution, and some birth–death chains.

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References

  1. Aldous, D., Diaconis, P.: Shuffling cards and stopping times. Am. Math. Monthly 93, 333–348 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aldous, D., Diaconis, P.: Strong uniform times and finite random walks. Adv. Appl. Math. 8, 69–97 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  3. Belsley, E.: Rates of convergence of random walk on distance regular graphs. Probab. Theory Relat. Fields 112, 493–533 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Borodin, A.: Multiplicative central measures in the Schur graph. In: Vershik, A.M. (ed.) Representation Theory, Dynamical Systems, Combinatorial and Algorithmic Methods II. Zap. Nauchn. Sem. POMI 240, Nauka, St Petersburg, 1997, 44–52 (Russian); English translation in J. Math. Sci. (New York) vol. 96, pp. 3472–3477 (1999)

  5. Borodin, A., Olshanski, G.: Harmonic functions on multiplicative graphs and interpolation polynomials. Electron. J. Combin. 7, Research paper 28, 39 p (electronic) (2000)

  6. Borodin, A., Olshanski, G.: Infinite dimensional diffusions as limits of random walks on partitions, arXiv: math.PR/0706.1034 (2007)

  7. Borodin, A., Olshanski, G.: Markov processes on partitions. Probab. Theory Relat. Fields 135, 84–152 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Borodin, A., Olshanski, G.: Z-measures on partitions and their scaling limits. Eur. J. Combin. 26, 795–834 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Brown, M.: Spectral analysis, without eigenvectors, for Markov chains. Probab. Eng. Inf. Sci. 5, 131–144 (1991)

    MATH  Google Scholar 

  10. Brown, M., Shao, Y.: Identifying coefficients in the spectral representation for first passage time distributions. Probab. Eng. Inf. Sci. 1, 69–74 (1987)

    Article  MATH  Google Scholar 

  11. Chatterjee, S.: Stein’s method for concentration inequalities. Probab. Theory Relat. Fields 138, 305–312 (2007)

    Article  MATH  Google Scholar 

  12. Chatterjee, S., Diaconis, P., Meckes, E.: Exchangeable pairs and Poisson approximation. Probab. Surv. 2, 64–106 (2005)

    Article  MathSciNet  Google Scholar 

  13. D’Aristotle, A.: The nearest neighbor random walk on subspaces of a vector space and rate of convergence. J. Theor. Probab. 8, 321–346 (1993)

    Article  Google Scholar 

  14. Diaconis, P.: The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93, 1659–1664 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  15. Diaconis, P., Fill, J.: Strong stationary times via a new form of duality. Ann. Probab. 18, 1483–1522 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. Diaconis, P., Hanlon, P.: Eigen-analysis for some examples of the Metropolis algorithm. In: Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, vol. 138, pp. 99–117, Contemp. Math. (1992)

  17. Diaconis, P., Saloff-Coste, L.: Separation cutoffs for birth death chains. Ann. Appl. Probab. 16, 2098–2122 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Diaconis, P., Shahshahani, M.: Time to reach stationarity in the Bernoulli–Laplace diffusion model. SIAM J. Math. Anal. 18, 208–218 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  19. Ewens, W.J.: Population genetics theory: the past and the future. In: Mathematical and Statistical Developments of Evolutionary Theory, pp. 117–228, Kluwer, Dordrecht (1990)

  20. Fomin, S.: Duality of graded graphs. J. Algebraic Combin. 3, 357–404 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  21. Fulman, J.: Stein’s method and Plancherel measure of the symmetric group. Trans. Am. Math. Soc. 357, 555–570 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fulman, J.: Convergence rates of random walk on irreducible representations of finite groups. J. Theor. Probab. 21, 193–211 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Fulman, J.: Separation cutoffs for random walk on irreducible representations, arXiv: math.PR/ 0703291 (2007)

  24. Fulman, J.: Stein’s method and random character ratios. Trans. Am. Math. Soc. (to appear)

  25. Fulman, J.: Stein’s method, Jack measure, and the Metropolis algorithm. J. Combin. Theory Ser. A. 108, 275–296 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hoffman, P., Humphreys, J.: Projective Representations of the Symmetric Group. Oxford University Press, New York (1992)

    Google Scholar 

  27. Ivanov, V.: Plancherel measure on shifted Young diagrams. In: Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, Am. Math. Soc. Transl. Ser. 2, vol. 217, pp. 73–86 (2006)

  28. Karlin, S., McGregor, J.: Ehrenfest urn models. J. Appl. Probab. 2, 352–376 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kerov, S.: The boundary of Young lattice and random Young tableaux. Formal power series and algebraic combinatorics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. vol. 24. Amer. Math. Soc., Providence, pp. 133–158 (1996)

  30. Kerov, S., Olshanski, G., Vershik, A.: Harmonic analysis on the infinite symmetric group. A deformation of the regular representation. C. R. Acad. Sci. Paris Sér. I Math. 316, 773–778 (1993)

    MATH  MathSciNet  Google Scholar 

  31. Kerov, S., Olshanski, G., Vershik, A.: Harmonic analysis on the infinite symmetric group. Invent. Math. 158, 551–642 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. Macdonald, I.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, New York (1995)

    MATH  Google Scholar 

  33. Matsumoto, S.: Correlation functions of the shifted Schur measure. J. Math. Soc. Jpn. 57, 619–637 (2005)

    Article  MATH  Google Scholar 

  34. Okounkov, A.: SL(2) and z measures. In: Random matrix models and their applications, Math. Sci. Res. Inst. Publ. vol. 40, pp. 407-420, Cambridge University Press, Cambridge (2001)

  35. Okounkov, A.: The uses of random partitions. In: XIVth International Congress on Mathematical Physics, pp. 379–403. World Scientific Publications, Hackensack (2005)

  36. Pak, I.: Random walk on groups: strong uniform time approach, PhD Thesis, Harvard University (1997)

  37. Petrov, L.: Two-parameter family of diffusion processes in the Kingman simplex, arXiv: math.PR/0708. 1930 (2007)

  38. Ross, N.: Step size in Stein’s method of exchangeable pairs (2007) (preprint)

  39. Sagan, B.: The Symmetric Group. Representations, Combinatorial Algorithms, and Symmetric Functions. Springer, New York (1991)

    MATH  Google Scholar 

  40. Saloff-Coste, L.: Random walk on finite groups. In: Probability on Discrete Structures, vol. 110, pp. 263-346, Encyclopedia Math. Sci. Springer, Berlin (2004)

  41. Stanley, R.: Enumerative Combinatorics, vol. 1. Wadsworth and Brooks/Cole, Monterey (1986)

    Google Scholar 

  42. Stanley, R.: Differential posets. J. Am. Math. Soc. 1, 919–961 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  43. Stanley, R.: Variations on differential posets. In: Invariant Theory and Tableaux, IMA Vol. Math. Appl. vol. 19, pp. 145–165, Springer, New York (1990)

  44. Tracy, C., Widom, H.: A limit theorem for shifted Schur measures. Duke Math. J. 123, 171–208 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  45. Wilf, H.: Generatingfunctionology, 2nd edn. Academic Press, Boston (1994)

    MATH  Google Scholar 

  46. Wilkinson, J.: The Algebraic Eigenvalue Problem. Oxford University Press, Oxford (1988)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Southern California, Los Angeles, CA, 90089, USA

    Jason Fulman

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  1. Jason Fulman
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Correspondence to Jason Fulman.

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Cite this article

Fulman, J. Commutation relations and Markov chains. Probab. Theory Relat. Fields 144, 99–136 (2009). https://doi.org/10.1007/s00440-008-0143-0

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  • Received: 09 December 2007

  • Revised: 21 January 2008

  • Published: 13 February 2008

  • Issue Date: May 2009

  • DOI: https://doi.org/10.1007/s00440-008-0143-0

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Keywords

  • Commutation relations
  • Separation distance
  • Differential poset
  • Markov chain
  • Symmetric function
  • Ewens distribution

Mathematics Subject Classification (2000)

  • 60J10
  • 60C05
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