Skip to main content

Advertisement

SpringerLink
Log in
Menu
Find a journal Publish with us
Search
Cart
  1. Home
  2. Probability Theory and Related Fields
  3. Article
Universality properties of Gelfand–Tsetlin patterns
Download PDF
Download PDF
  • Published: 06 December 2011

Universality properties of Gelfand–Tsetlin patterns

  • Anthony P. Metcalfe1 

Probability Theory and Related Fields volume 155, pages 303–346 (2013)Cite this article

Abstract

A standard Gelfand–Tsetlin pattern of depth n is a configuration of particles in \({\{1,\ldots,n\} \times \mathbb{R}}\) . For each \({r \in \{1, \ldots, n\}, \{r\} \times \mathbb{R}}\) is referred to as the rth level of the pattern. A standard Gelfand–Tsetlin pattern has exactly r particles on each level r, and particles on adjacent levels satisfy an interlacing constraint. Probability distributions on the set of Gelfand–Tsetlin patterns of depth n arise naturally as distributions of eigenvalue minor processes of random Hermitian matrices of size n. We consider such probability spaces when the distribution of the matrix is unitarily invariant, prove a determinantal structure for a broad subclass, and calculate the correlation kernel. In particular we consider the case where the eigenvalues of the random matrix are fixed. This corresponds to choosing uniformly from the set of Gelfand–Tsetlin patterns whose nth level is fixed at the eigenvalues of the matrix. Fixing \({q_n \in \{1,\ldots,n\}}\) , and letting n → ∞ under the assumption that \({\frac{q_n}n \to \alpha \in (0, 1)}\) and the empirical distribution of the particles on the nth level converges weakly, the asymptotic behaviour of particles on level q n is relevant to free probability theory. Saddle point analysis is used to identify the set in which these particles behave asymptotically like a determinantal random point field with the Sine kernel.

Download to read the full article text

Working on a manuscript?

Avoid the common mistakes

References

  1. Anderson G., Guionnet A., Zeitouni O.: An Introduction to Random Matrices. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  2. Baryshnikov Y.: GUEs and queues. Probab. Theory Related Fields. 119, 256–274 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Belinschi S.: The atoms of the free multiplicative convolution of two probability distributions. Integral Equ. Oper. Theory. 46(4), 377–386 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belinschi, S.: Complex analysis methods in noncommutative probability. Ph.D. thesis, Indiana University, Bloomington (2006)

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Collins, B.: Intégrales matricielles et probabilités non-commutatives. Thése de doctorat de lUniversité Paris 6, (2003)

  7. Collins B.: Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral and free probability. IMRN 17, 953–982 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Erdös L.: Universality of Wigner random matrices: a survey of recent results. Russ. Math. Surv 66(3), 507–626 (2011)

    Article  MATH  Google Scholar 

  10. Ferrari P., Spohn H.: Step fluctuations for a faceted crystal. J. Stat. Phys. 113, 1–46 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fleming B., Forrester P., Nordenstam E.: A finitization of the bead process. Probab. Theory Relat. Fields 1, 36 (2010)

    Google Scholar 

  12. Forrester, P., Nagao, T.: Determinantal correlations for classical projection processes. J. Stat. Mech. P08011 (2011)

  13. Fyodorov Y., Strahov E.: An exact formula for general spectral correlation function of random Hermitian matrices. J. Phys. A Math. Gen. 36, 3203–3214 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Horn R., Johnson C.: Matrix Analysis. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  15. Johansson K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215, 683–705 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Johansson K.: Discrete polynuclear growth and determinantal processes. Commun. Math. Phys. 242, 277–329 (2003)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Johansson, K.: Random matrices and determinantal processes. Math. stat. phys., session LXXXIII. In: Lecture Notes of the Les Houches Summer School. Elsevier Science, Amsterdam, pp. 1–56 (2006)

  19. Defosseux M.: Orbit measures, random matrix theory and interlaced determinantal processes. Ann. Inst. H. Poincaré Probab. Stat 46(1), 209–249 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mehta M.: Random Matrices. Elsevier, Amsterdam (2004)

    MATH  Google Scholar 

  21. Metcalfe A., O’Connell N., Warren J.: Interlaced processes on the circle. Ann. Inst. H. Poincaré Probab. Stat. 45(4), 1165–1184 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Nica, A., Speicher, R.: Lectures on the combinatorics of free probability. Cambridge University Press (2006)

  23. Nordenstam, E.: Interlaced particles in tilings and random matrices. KTH. Ph.D. thesis (2009)

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

    MathSciNet  MATH  Google Scholar 

  25. Pastur L.A., Shcherbina M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Stat. Phys. 86, 109–147 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rudin W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)

    MATH  Google Scholar 

  27. Soshnikov A.: Determinantal random point fields. Russ. Math. Surv. 55, 923–975 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Szegö G.: Orthogonal Polynomials. American Mathematical Society, Providence (1939)

    Google Scholar 

  29. Voiculescu D.: A strengthened asymptotic freeness result for random matrices with applications to free entropy. Internat. Math. Res. Notices. 1, 41–63 (1998)

    Article  MathSciNet  Google Scholar 

  30. Xu F.: A random matrix model from two-dimensional Yang-Mills theory. Commun. Math. Phys. 190(2), 287–307 (1997)

    Article  MATH  Google Scholar 

  31. Warren J.: Dyson’s Brownian motions, intertwining and interlacing. Electron. J. Probab. 12(19), 573–590 (2007)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Anthony P. Metcalfe

Authors
  1. Anthony P. Metcalfe
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anthony P. Metcalfe.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Metcalfe, A.P. Universality properties of Gelfand–Tsetlin patterns. Probab. Theory Relat. Fields 155, 303–346 (2013). https://doi.org/10.1007/s00440-011-0399-7

Download citation

  • Received: 17 May 2011

  • Revised: 15 September 2011

  • Published: 06 December 2011

  • Issue Date: February 2013

  • DOI: https://doi.org/10.1007/s00440-011-0399-7

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Mathematics Subject Classification (2000)

  • 60B20
Download PDF

Working on a manuscript?

Avoid the common mistakes

Advertisement

Search

Navigation

  • Find a journal
  • Publish with us

Discover content

  • Journals A-Z
  • Books A-Z

Publish with us

  • Publish your research
  • Open access publishing

Products and services

  • Our products
  • Librarians
  • Societies
  • Partners and advertisers

Our imprints

  • Springer
  • Nature Portfolio
  • BMC
  • Palgrave Macmillan
  • Apress
  • Your US state privacy rights
  • Accessibility statement
  • Terms and conditions
  • Privacy policy
  • Help and support

167.114.118.210

Not affiliated

Springer Nature

© 2023 Springer Nature