Extreme gaps between eigenvalues of random matrices

Annals of Probability

Volumn 41, Issue 4, 2013, Pages 2648-2681

  Arous, Gérard Ben ^a   Bourgade, Paul ^b  

^a NEW YORK UNIVERSITY   (United States)

^b HARVARD UNIVERSITY   (United States)

Author keywords

Eigenvalues statistics; Extreme spacings; Gaussian unitary ensemble; Negative association property; Random matrices

Indexed keywords

EID: 84881529077     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/11-AOP710     Document Type: Article

References (35)

1. Anderson, G. W., Guionnet, A. and Zeitouni, O. (2010). An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics 118. Cambridge Univ. Press, Cambridge. MR2760897
2. Ben Arous, G. and Guionnet, A. (2011). Wigner matrices, Chapter 21. In The Oxford Handbook on Random Matrix Theory (G. Akemann, J. Baik and P. Di Francesco, eds.) 433-451. Oxford Univ. Press, Oxford.
3. Ben Arous, G. and Kuptsov, A. (2009). REM universality for random Hamiltonians. In Spin Glasses: Statics and Dynamics. Progress in Probability 62 45-84. Birkhäuser, Basel. MR2761980
4. Borcea, J., Brändén, P. and Liggett, T. M. (2009). Negative dependence and the geometry of polynomials. J. Amer. Math. Soc. 22 521-567. MR2476782
5. Bui, H. M. (2011). Large gaps between consecutive zeros of the Riemann zeta-function. J. Number Theory 131 67-95. MR2729210
6. Deift, P. (2008). Some open problems in random matrix theory and the theory of integrable systems. In Integrable Systems and Random Matrices. Contemporary Mathematics 458 419-430. Amer. Math. Soc., Providence, RI. MR2411922
7. 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
8. Deift, P., Kriecherbauer, T., Mclaughlin, K. T. R., Venakides, S. and Zhou, X. (1999). Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52 1335-1425. MR1702716
9. Deift, P., Nanda, T. and Tomei, C. (1983). Ordinary differential equations and the symmetric eigenvalue problem. SIAM J. Numer. Anal. 20 1-22. MR0687364
10. Delyon, B. and Yao, J. (2006). On the spectral distribution of Gaussian random matrices. Acta Math. Appl. Sin. Engl. Ser. 22 297-312. MR2216482
11. Diaconis, P. (2003). Patterns in eigenvalues: The 70th Josiah Willard Gibbs lecture. Bull. Amer. Math. Soc. (N.S.) 40 155-178. MR1962294
12. Dyson, F. J. (1962). Statistical theory of the energy levels of complex systems. I. J. Math. Phys. 3 140-156. MR0143556
13. Ehrhardt, T. (2006). Dyson's constant in the asymptotics of the Fredholm determinant of the sine kernel. Comm. Math. Phys. 262 317-341. MR2200263
14. Erdo S, L., Schlein, B. and Yau, H.-T. (2011). Universality of random matrices and local relaxation flow. Invent. Math. 185 75-119. MR2810797
15. Georgii, H.-O. and Yoo, H. J. (2005). Conditional intensity and Gibbsianness of determinantal point processes. J. Stat. Phys. 118 55-84. MR2122549
16. Gohberg, I., Goldberg, S. and Krupnik, N. (2000). Traces and Determinants of Linear Operators. Operator Theory: Advances and Applications 116. Birkhäuser, Basel. MR1744872
17. Gourdon, X. (2004). Computation of zeros of the Zeta function. Unpublished manuscript.
18. Horn, R. A. and Johnson, C. R. (1985). Matrix Analysis. Cambridge Univ. Press, Cambridge. MR0832183
19. Johansson, K. (2005). Random matrices and determinantal processes. Lecture notes, Summer School on Mathematical Statistical Mechanics at Ecole de Physique, Les Houches.
20. Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie Verlag, Berlin. MR0818219
21. Katz, N. M. and Sarnak, P. (1999). Random Matrices, Frobenius Eigenvalues, and Monodromy. American Mathematical Society Colloquium Publications 45. Amer. Math. Soc., Providence, RI. MR1659828
22. Keating, J. P. and Snaith, N. C. (2000). Random matrix theory and (1/2 + it). Comm. Math. Phys. 214 57-89. MR1794265
23. 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. IMRN 25 1249-1272. MR2047176
24. Lyons, R. (2003). Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. 98 167-212. MR2031202
25. Mehta, M. L. (2004). Random Matrices, 3rd ed. Pure and Applied Mathematics (Amsterdam) 142. Elsevier, Amsterdam. MR2129906
26. Montgomery, H. L. (1973). The pair correlation of zeros of the zeta function. In Analytic Number Theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972) 181-193. Amer. Math. Soc., Providence, RI. MR0337821
27. Odlyzko, A. M. (1987). On the distribution of spacings between zeros of the zeta function. Math. Comp. 48 273-308. MR0866115
28. Shirai, T. and Takahashi, Y. (2003). Random point fields associated with certain Fredholm determinants. II. Fermion shifts and their ergodic and Gibbs properties. Ann. Probab. 31 1533-1564. MR1989442
29. Soshnikov, A. (1998). Level spacings distribution for large random matrices: Gaussian fluctuations. Ann. of Math. (2) 148 573-617. MR1668559
30. Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55 923-975.
31. Soshnikov, A. (2005). Statistics of extreme spacing in determinantal random point processes. Mosc. Math. J. 5 705-719, 744. MR2241818
32. Szego G. (1975). Orthogonal Polynomials, 4th ed. American Mathematical Society Colloquium Publications XXIII. Amer. Math. Soc., Providence, RI. MR0372517
33. Tao, T. and Vu, V. (2011). Random matrices: Universality of local eigenvalue statistics. Acta Math. 206 127-204. MR2784665
34. Vinson, J. (2001). Closest spacing of eigenvalues. Ph. D. thesis, Princeton Univ.
35. Widom, H. (1971). The strong Szego limit theorem for circular arcs. Indiana Univ. Math. J. 21 277-283. MR0288495