Large gaps between random eigenvalues

Annals of Probability

Volumn 38, Issue 3, 2010, Pages 1263-1279

  Valkó, Benedek ^a   Virág, Bálint ^b  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b UNIVERSITY OF TORONTO   (Canada)

Author keywords

Ensembles; Eigenvalues of random matrices; Large deviation

Indexed keywords

EID: 77957037478     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/09-AOP508     Document Type: Article

References (19)

1. BASOR, E. L., TRACY, C. A. and WIDOM, H. (1992). Asymptotics of level-spacing distributions for random matrices. Phys. Rev. Lett. 69 5-8. MR1173848.
2. CAMBRONERO, S., RIDER, B. and RAMÍREZ, J. (2006). On the shape of the ground state eigenvalue density of a random Hill's equation. Comm. Pure Appl. Math. 59 935-976. MR2222441.
3. CHEN, Y. and MANNING, S. M. (1996). Some eigenvalue distribution functions of the Laguerre ensemble. J. Phys. A 29 7561-7579. MR1425838.
4. DEIFT, P., ITS, A., KRASOVSKY, I. and ZHOU, X. (2007). The Widom-Dyson constant for the gap probability in random matrix theory. J. Comput. Appl. Math. 202 26-47. MR2301810.
5. DEIFT, P. A., ITS, A. R. and ZHOU, X. (1997). A Riemann-Hilbert approach to asymptotic problems arising in the theory of random matrix models, and also in the theory of integrable statistical mechanics. Ann. of Math. (2) 146 149-235. MR1469319.
6. DES CLOIZEAUX, J. and MEHTA, M. L. (1973). Asymptotic behavior of spacing distributions for the eigenvalues of random matrices. J. Math. Phys. 14 1648-1650. MR0328158.
7. DYSON, F. J. (1962). Statistical theory of the energy levels of complex systems. II. J. Math. Phys. 3 157-165. MR0143557.
8. EHRHARDT, T. (2006). Dyson's constant in the asymptotics of the Fredholm determinant of the sine kernel. Comm. Math. Phys. 262 317-341. MR2200263.
9. KARATZAS, I. and SHREVE, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York. MR1121940.
10. KRASOVSKY, I. V. (2004). Gap probability in the spectrum of random matrices and asymptotics of polynomials orthogonal on an arc of the unit circle. Int. Math. Res. Not. 25 1249-1272. MR2047176.
11. MCKEAN, H. P. (2005). Stochastic Integrals. AMS Chelsea Publishing, Providence, RI. MR2169626.
12. RAMÍREZ, J., RIDER, B. and VIRÁG, B. (2007). Beta ensembles, stochastic Airy spectrum, and a diffusion. Available at math/0607331.
13. REVUZ, D. and YOR, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin. MR1725357.
14. TRACY, C. A. and WIDOM, H. (1993). Introduction to random matrices. In Geometric and Quantum Aspects of Integrable Systems (Scheveningen, 1992). Lecture Notes in Physics 424 103-130. Springer, Berlin. MR1253763.
15. VALKÓ, B. and VIRÁG, B. (2009). Continuum limits of random matrices and the Brownian carousel. Invent. Math. 177 463-508. MR2534097.
16. WIDOM, H. (1994). The asymptotics of a continuous analogue of orthogonal polynomials. J. Approx. Theory 77 51-64. MR1273699.
17. WIGNER, E. P. (1951). On a class of analytic functions from the quantum theory of collisions. Ann. of Math. (2) 53 36-67. MR0039059.
18. WIGNER, E. P. (1952). On the connection between the distribution of poles and residues for an R function and its invariant derivative. Ann. of Math. (2) 55 7-18. MR0046433.
19. WIGNER, E. P. (1957). Gatlinburg Conference on Neutron Physics. Report 2309:59, Oak Ridge National Laboratory, Oak Ridge, TN.