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A finitization of the bead process
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  • Published: 20 October 2010

A finitization of the bead process

  • Benjamin J. Fleming1,
  • Peter J. Forrester1 &
  • Eric Nordenstam2 

Probability Theory and Related Fields volume 152, pages 321–356 (2012)Cite this article

Abstract

The bead process is the particle system defined on parallel lines, with underlying measure giving constant weight to all configurations in which particles on neighbouring lines interlace, and zero weight otherwise. Motivated by the statistical mechanical model of the tiling of an abc-hexagon by three species of rhombi, a finitized version of the bead process is defined. The corresponding joint distribution can be realized as an eigenvalue probability density function for a sequence of random matrices. The finitized bead process is determinantal, and we give the correlation kernel in terms of Jacobi polynomials. Two scaling limits are considered: a global limit in which the spacing between lines goes to zero, and a certain bulk scaling limit. In the global limit the shape of the support of the particles is determined, while in the bulk scaling limit the bead process kernel of Boutillier is reclaimed, after appropriate identification of the anisotropy parameter therein.

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References

  1. Baryshnikov Yu.: GUEs and queues. Probab. Theory Relat. Fields 119(2), 256–274 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baik J., Borodin A., Deift P., Suidan T.: A model for the bus system in Cuernavaca (Mexico). J. Phys. A 39(28), 8965–8975 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Borodin A., Ferrari P.L., Prähofer M., Sasamoto T.: Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129(5–6), 1055–1080 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bosbach, C., Gawronski, W.: Strong asymptotics for Jacobi polynomials with varying weights. Methods Appl. Anal. 6(1), 39–54 (1999). Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part I

  5. Boutillier C.: The bead model and limit behaviors of dimer models. Ann. Probab. 37(1), 107–142 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Borodin A., Rains E.M.: Eynard-Mehta theorem, Schur process, and their Pfaffian analogs. J. Stat. Phys. 121(3–4), 291–317 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen L.-C., Ismail M.E.H.: On asymptotics of Jacobi polynomials. SIAM J. Math. Anal. 22(5), 1442–1449 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  8. Collins B.: Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Relat. Fields 133(3), 315–344 (2005)

    Article  MATH  Google Scholar 

  9. Forrester, P.J., Nagao, T.: Determinantal correlations for classical projection processes. (2008). arXiv:0801.0100

  10. Forrester P.J., Nordenstam E.: The anti-symmetric GUE minor process. Mosc. Math. J. 9(4), 749–774, 934 (2009)

    MATH  MathSciNet  Google Scholar 

  11. Forrester P.J., Nagao T., Honner G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges. Nuclear Phys. B 553(3), 601–643 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Forrester, P.J.: Log-gases and random matrices. In: London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton (2010)

  13. Forrester P.J., Rains E.M.: Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter. Probab. Theory Relat. Fields 130(4), 518–576 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  14. Forrester P.J., Rains E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Probab. Theory Relat. Fields 131(1), 1–61 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gorin V.E.: Nonintersecting paths and the Hahn orthogonal polynomial ensemble. Funktsional. Anal. i Prilozhen. 42(3), 23–44, 96 (2008)

    Article  MathSciNet  Google Scholar 

  16. Gawronski, W., Shawyer, B.: Strong asymptotics and the limit distribution of the zeros of Jacobi polynomials \({P_n^{(an+\alpha,bn+\beta)}}\). In: Progress in Approximation Theory, pp. 379–404. Academic Press, Boston (1991)

  17. Izen S.H.: Refined estimates on the growth rate of Jacobi polynomials. J. Approx. Theory 144(1), 54–66 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Johansson K., Nordenstam E.: Eigenvalues of GUE minors. Electron. J. Probab. 11(50), 1342–1371 (2006)

    MATH  MathSciNet  Google Scholar 

  19. Johansson K.: Non-intersecting paths, random tilings and random matrices. Probab. Theory Relat. Fields 123(2), 225–280 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  20. Nagao T., Forrester P.J.: Multilevel dynamical correlation functions for Dyson’s Brownian motion model of random matrices. Phys. Lett. A 247(1–2), 42–46 (1998)

    Article  Google Scholar 

  21. Nordenstam, E.: Interlaced particles in tilings and random matrices. PhD thesis, Swedish Royal Institute of Technology (KTH) (2009)

  22. Okounkov A., Reshetikhin N.: The birth of a random matrix. Mosc. Math. J. 6(3), 553–566, 588 (2006)

    MATH  MathSciNet  Google Scholar 

  23. Szegő, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence (1975). American Mathematical Society, Colloquium Publications, vol. XXIII

  24. Wachter K.W.: The limiting empirical measure of multiple discriminant ratios. Ann. Stat. 8(5), 937–957 (1980)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, The University of Melbourne, Victoria, 3010, Australia

    Benjamin J. Fleming & Peter J. Forrester

  2. Institutionen för Matematik, Swedish Royal Institute of Technology (KTH), 100 44, Stockholm, Sweden

    Eric Nordenstam

Authors
  1. Benjamin J. Fleming
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  2. Peter J. Forrester
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  3. Eric Nordenstam
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Correspondence to Eric Nordenstam.

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Fleming, B.J., Forrester, P.J. & Nordenstam, E. A finitization of the bead process. Probab. Theory Relat. Fields 152, 321–356 (2012). https://doi.org/10.1007/s00440-010-0324-5

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  • Received: 06 July 2009

  • Revised: 09 September 2010

  • Published: 20 October 2010

  • Issue Date: February 2012

  • DOI: https://doi.org/10.1007/s00440-010-0324-5

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Mathematics Subject Classification (2010)

  • 60G55
  • 60C05
  • 60B20
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