Thinning and conditioning of the circular unitary ensemble. (English) Zbl 1386.60025
Summary: We apply the operation of random independent thinning on the eigenvalues of \(n\times n\) Haar distributed unitary random matrices. We study gap probabilities for the thinned eigenvalues, and we study the statistics of the eigenvalues of random unitary matrices which are conditioned such that there are no thinned eigenvalues on a given arc of the unit circle. Various probabilistic quantities can be expressed in terms of Toeplitz determinants and orthogonal polynomials on the unit circle, and we use these expressions to obtain asymptotics as \(n\to\infty\).
MSC:
| 60B20 | Random matrices (probabilistic aspects) |
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 35Q15 | Riemann-Hilbert problems in context of PDEs |
Keywords:
random matrix theory; thinning of point processes; Toeplitz determinants; Riemann-Hilbert problems; orthogonal polynomialsSoftware:
DLMFReferences:
| [1] | Anderson, G. W., Guionnet, A. and Zeitouni, O., An Introduction to Random Matrices, , Vol. 118 (Cambridge University Press, Cambridge, 2010). · Zbl 1184.15023 |
| [2] | Baik, J., Deift, P. and Johansson, K., On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc.12 (1999) 1119-1178. · Zbl 0932.05001 |
| [3] | Basor, E., Asymptotic formulas for Toeplitz determinants, Trans. Amer. Math. Soc.239 (1978) 33-65. · Zbl 0409.47018 |
| [4] | Bleher, P., Lectures on random matrix models, The Riemann-Hilbert approach, in Random Matrices, Random Processes and Integrable Systems, (Springer, New York, 2011), pp. 251-349. · Zbl 1252.15040 |
| [5] | Bohigas, O. and Pato, M. P., Missing levels in correlated spectra, Phys. Lett. B595 (2004) 171-176. |
| [6] | Bohigas, O. and Pato, M. P., Randomly incomplete spectra and intermediate statistics, Phys. Rev. E (3)74 (2006) 036212. |
| [7] | Bothner, T., Deift, P., Its, A. and Krasovsky, I., On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I, Comm. Math. Phys.337(3) (2015) 1397-1463. · Zbl 1321.82027 |
| [8] | Böttcher, A. and Silbermann, B., Toeplitz operators and determinants generated by symbols with one Fisher-Hartwig singularity, Math. Nachr.127 (1986) 95-123. · Zbl 0613.47024 |
| [9] | Charlier, C. and Claeys, T., Asymptotics for Toeplitz determinants: Perturbation of symbols with a gap, J. Math. Phys.56 (2015) 022705. · Zbl 1310.15051 |
| [10] | Claeys, T. and Fahs, B., Random matrices with merging singularities and the Painlevé V equation, SIGMA12 (2016) 031. · Zbl 1338.60013 |
| [11] | Claeys, T. and Krasovsky, I., Toeplitz determinants with merging singularities, Duke Math. J.164(15) (2015) 2897-2987. · Zbl 1333.15018 |
| [12] | Daley, D. J. and Vere-Jones, D., An Introduction to the Theory of Point Processes: Volume II: General Theory and Structure, 2nd edn., (Springer, New York, 2008). · Zbl 1159.60003 |
| [13] | Deift, P., Its, A. and Krasovky, I., Asymptotics of Toeplitz, Hankel, and Toeplitz \(\)+ Hankel determinants with Fisher-Hartwig singularities, Ann. of Math.174 (2011) 1243-1299. · Zbl 1232.15006 |
| [14] | Deift, P., Its, A. and Krasovky, I., On the asymptotics of a Toeplitz determinant with singularities, MSRI Publ.65 (2014) 93-146. · Zbl 1326.35218 |
| [15] | Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X., 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 (1999) 1335-1425. · Zbl 0944.42013 |
| [16] | Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X., Strong asymptotics of orthogonal polynomials with respect to exponential weights, Comm. Pure Appl. Math.52 (1999) 1491-1552. · Zbl 1026.42024 |
| [17] | Deift, P. and Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math.137 (1993) 295-368. · Zbl 0771.35042 |
| [18] | Ehrhardt, T., A status report on the asymptotic behavior of Toeplitz determinants with Fisher-Hartwig singularities, Oper. Theory: Adv. Appl.124 (2001) 217-241. · Zbl 0993.47028 |
| [19] | Fisher, M. E. and Hartwig, R. E., Toeplitz determinants: Some applications, theorems, and conjectures, Adv. Chem. Phys.15 (1968) 333-353. |
| [20] | Fokas, A. S., Its, A. R. and Kitaev, A. V., The isomonodromy approach to matrix models in 2D quantum gravity, Comm. Math. Phys.147(2) (1992) 395-430. · Zbl 0760.35051 |
| [21] | Forrester, P. J. and Mays, A., Finite size corrections in random matrix theory and Odlyzko’s data set for the Riemann zeros, Proc. Roy. Soc. A471(2182) (2015), Article ID: 20150436, 21 pp. · Zbl 1371.11130 |
| [22] | Jimbo, M., Miwa, T., Môri, T. and Sato, M., Density matrix of an impenetrable Bose gas and the fifth Painlevé transcendent, Physica D1 (1980) 80-158. · Zbl 1194.82007 |
| [23] | Johansson, K., Random Matrices and Determinantal Processes, (Elsevier B.V., Amsterdam, 2006), pp. 1-55. · Zbl 1411.60144 |
| [24] | Kallenberg, O., A limit theorem for thinning of point processes, Inst. Stat. Mimeo Ser.908 (1974). |
| [25] | Keating, J. P. and Snaith, N. C., Random matrix theory and \(\zeta(1 / 2 + i t)\), Comm. Math. Phys.214 (2001) 57-89. · Zbl 1051.11048 |
| [26] | Kuijlaars, A. B. J., McLaughlin, K. T.-R., Van Assche, W. and Vanlessen, M., The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on \([- 1, 1]\), Adv. Math.188 (2004) 337-398. · Zbl 1082.42017 |
| [27] | Lavancier, F., Moller, J. and Rubak, E., Determinantal point process models and statistical inference: Extended version, J. Roy. Stat. Soc. Ser. B77(4) (2015) 853-877. · Zbl 1414.62403 |
| [28] | Mehta, M. L., Random Matrices, 3rd edn., , Vol. 142 (Elsevier Academic Press, 2004). · Zbl 1107.15019 |
| [29] | Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W., NIST Handbook of Mathematical Functions (, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010). · Zbl 1198.00002 |
| [30] | A. Rényi, A characterization of Poisson processes, in Collected Papers of Alfred Rényi (Academic Press, New York, 1956), pp. 519-527. · Zbl 0103.11503 |
| [31] | Saff, E. B. and Totik, V., Logarithmic Potentials with External Fields (Springer-Verlag, Berlin, 1997). · Zbl 0881.31001 |
| [32] | Simon, B., Orthogonal Polynomials on the Unit Circle, , Vol. 54 (American Mathematical Society, Providence, RI, 2005). · Zbl 1082.42020 |
| [33] | Soshnikov, A., Determinantal random point fields, Russian Math. Surveys55(5) (2000) 923-975. · Zbl 0991.60038 |
| [34] | Widom, H., The strong Szegő limit theorem for circular arcs, Indiana Univ. Math. J.21 (1971) 277-283. · Zbl 0213.34903 |
| [35] | Widom, H., Toeplitz determinants with singular generating functions, Amer. J. Math.95 (1973) 333-383. · Zbl 0275.45006 |
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