## Electronic Journal of Probability

### Eigenvalues of GUE Minors

#### Abstract

Consider an infinite random matrix $H=(h_{ij})_{0 < i,j}$ picked from the Gaussian Unitary Ensemble (GUE). Denote its main minors by $H_i=(h_{rs})_{1\leq r,s\leq i}$ and let the $j$:th largest eigenvalue of $H_i$ be $\mu^i_j$. We show that the configuration of all these eigenvalues $(i,\mu_j^i)$ form a determinantal point process on $\mathbb{N}\times\mathbb{R}$.

Furthermore we show that this process can be obtained as the scaling limit in random tilings of the Aztec diamond close to the boundary. We also discuss the corresponding limit for random lozenge tilings of a hexagon.

An Erratum to this paper has been published in Electronic Journal of Probability, Volume 12 (2007), paper number 37.

#### Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 50, 1342-1371.

Dates
Accepted: 20 December 2006
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464730585

Digital Object Identifier
doi:10.1214/EJP.v11-370

Mathematical Reviews number (MathSciNet)
MR2268547

Zentralblatt MATH identifier
1127.60047

Subjects
Primary: 60G55: Point processes
Secondary: 15A52 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20]

Keywords
Random matrices Tiling problems

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

#### Citation

Johansson, Kurt; Nordenstam, Eric. Eigenvalues of GUE Minors. Electron. J. Probab. 11 (2006), paper no. 50, 1342--1371. doi:10.1214/EJP.v11-370. https://projecteuclid.org/euclid.ejp/1464730585

#### References

• Baryshnikov, Yu. GUEs and queues. Probab. Theory Related Fields 119 (2001), no. 2, 256–274.
• Borodin, Alexei. Duality of orthogonal polynomials on a finite set. J. Statist. Phys. 109 (2002), no. 5-6, 1109–1120.
• Cohn, Henry; Larsen, Michael; Propp, James. The shape of a typical boxed plane partition. New York J. Math. 4 (1998), 137–165 (electronic).
• Daley, D. J.; Vere-Jones, D. An introduction to the theory of point processes. Springer Series in Statistics. Springer-Verlag, New York, 1988. xxii+702 pp. ISBN: 0-387-96666-8
• Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James. Alternating-sign matrices and domino tilings. I. J. Algebraic Combin. 1 (1992), no. 2, 111–132.
• Elkies, Noam; Kuperberg, Greg; Larsen, Michael; Propp, James. Alternating-sign matrices and domino tilings. II. J. Algebraic Combin. 1 (1992), no. 3, 219–234.
• Johansson, Kurt. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000), no. 2, 437–476.
• Johansson, Kurt. Discrete polynuclear growth and determinantal processes. Comm. Math. Phys. 242 (2003), no. 1-2, 277–329.
• Johansson, Kurt. The arctic circle boundary and the Airy process. Ann. Probab. 33 (2005), no. 1, 1–30.
• Johansson, Kurt. Non-intersecting, simple, symmetric random walks and the extended Hahn kernel. Ann. Inst. Fourier (Grenoble) 55 (2005), no. 6, 2129–2145.
• Johansson, Kurt Random matrices and determinantal processes. (2005) arXiv:math-ph/051003.
• Jockusch, William; Propp, James; Shor, Peter. Random domino tilings and the arctic circle theorem. (1998) arXiv:math.CO/9801068.
• Koekoek, Roelof;Swarttouw, Rene F. The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Technical Report DUT-TWI-98-17, Delft University of Technology, Delft, The Netherlands, 1998. Available at http://citeseer.nj.nec.com/62227.html.
• Mehta, Madan Lal. Random matrices. Second edition. Academic Press, Inc., Boston, MA, 1991. xviii+562 pp. ISBN: 0-12-488051-7.
• Okounkov, Andrei; Reshetikhin, Nikolai. Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16 (2003), no. 3, 581–603 (electronic).
• Okounkov, Andrei; Reshetikhin, Nikolai. The birth of a random matrix, (2006) Preprint.
• Propp, James. Generalized domino-shuffling. Tilings of the plane. Theoret. Comput. Sci. 303 (2003), no. 2-3, 267–301.
• Sagan, Bruce E. The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, 2001. xvi+238 pp. ISBN: 0-387-95067-2.
• Soshnikov, A. Determinantal random point fields. (Russian) Uspekhi Mat. Nauk 55 (2000), no. 5(335), 107–160; translation in Russian Math. Surveys 55 (2000), no. 5, 923–975.
• Stanley, Richard P. Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. xii+581 pp. ISBN: 0-521-56069-1; 0-521-78987-7.