Asymptotics of Fredholm determinant associated with the Pearcey kernel. (English) Zbl 1469.60027
The Pearcey kernel is a classical and universal kernel arising from random matrix theory, which describes the local statistics of eigenvalues when the limiting mean eigenvalue density exhibits a cusp-like singularity. It appears in a variety of statistical physics models beyond matrix models as well. The authors consider the Fredholm determinant of a trace class operator acting on \(L^2 (s,s)\) with the Pearcey kernel. Based on a steepest descent analysis for a \(3 \times 3\) matrix-valued Riemann-Hilbert problem, it is obtained asymptotics of the Fredholm determinant as \(s \rightarrow +\infty,\) which is also interpreted as large gap asymptotics in the context of random matrix theory.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)
MSC:
| 60B20 | Random matrices (probabilistic aspects) |
| 15B52 | Random matrices (algebraic aspects) |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
Software:
DLMFReferences:
| [1] | Adler, M.; Cafasso, M.; van Moerbeke, P., From the Pearcey to the Airy process, Electron. J. Probab., 16, 1048-1064 (2011) · Zbl 1231.60107 · doi:10.1214/EJP.v16-898 |
| [2] | Adler, M.; Orantin, N.; van Moerbeke, P., Universality for the Pearcey process, Phys. D, 239, 924-941 (2010) · Zbl 1189.82085 · doi:10.1016/j.physd.2010.01.005 |
| [3] | Adler, M.; van Moerbeke, P., PDEs for the Gaussian ensemble with external source and the Pearcey distribution, Commun. Pure Appl. Math., 60, 1261-1292 (2007) · Zbl 1193.82012 · doi:10.1002/cpa.20175 |
| [4] | Ajanki, OH; Erdős, L.; Krüger, T., Singularities of solutions to quadratic vector equations on the complex upper half-plane, Commun. Pure Appl. Math., 70, 1672-1705 (2017) · Zbl 1419.15014 · doi:10.1002/cpa.21639 |
| [5] | Alt, J.; Erdős, L.; Krüger, T., The Dyson equation with linear self-energy: spectral bands, edges and cusps, Doc. Math., 25, 1421-1539 (2020) · Zbl 1450.60005 |
| [6] | Baik, J.; Buckingham, R.; DiFranco, J., Asymptotics of Tracy-Widom distributions and the total integral of a Painlevé II function, Commun. Math. Phys., 280, 463-497 (2008) · Zbl 1221.33032 · doi:10.1007/s00220-008-0433-5 |
| [7] | Basor, EL; Ehrhardt, T., On the asymptotics of certain Wiener-Hopf-plus-Hankel determinants, N. Y. J. Math., 11, 171-203 (2005) · Zbl 1099.47026 |
| [8] | Bertola, M.; Cafasso, M., The transition between the gap probabilities from the Pearcey to the Airy process-a Riemann-Hilbert approach, Int. Math. Res. Not. IMRN, 2012, 1519-1568 (2012) · Zbl 1251.60005 · doi:10.1093/imrn/rnr066 |
| [9] | Bleher, PM; Kuijlaars, ABJ, Large \(n\) limit of Gaussian random matrices with external source, part III: double scaling limit, Commun. Math. Phys., 270, 481-517 (2007) · Zbl 1126.82010 · doi:10.1007/s00220-006-0159-1 |
| [10] | Borodin, A.; Deift, P., Fredholm determinants, Jimbo-Miwa-Ueno \(\tau \)-functions, and representation theory, Commun. Pure Appl. Math., 55, 1160-1230 (2002) · Zbl 1033.34089 · doi:10.1002/cpa.10042 |
| [11] | Brézin, E.; Hikami, S., Level spacing of random matrices in an external source, Phys. Rev. E., 58, 7176-7185 (1998) · doi:10.1103/PhysRevE.58.7176 |
| [12] | Brézin, E.; Hikami, S., Universal singularity at the closure of a gap in a random matrix theory, Phys. Rev. E., 57, 4140-4149 (1998) · doi:10.1103/PhysRevE.57.4140 |
| [13] | Brézin, E.; Hikami, S., Extension of level-spacing universality, Phys. Rev. E, 56, 264-269 (1997) · doi:10.1103/PhysRevE.56.264 |
| [14] | Brézin, E.; Hikami, S., Spectral form factor in a random matrix theory, Phys. Rev. E, 55, 4067-4083 (1997) · doi:10.1103/PhysRevE.55.4067 |
| [15] | Brézin, E.; Hikami, S., Correlations of nearby levels induced by a random potential, Nucl. Phys. B, 479, 697-706 (1996) · Zbl 0925.82117 · doi:10.1016/0550-3213(96)00394-X |
| [16] | Chen, Y.; Eriksen, K.; Tracy, CA, Largest eigenvalue distribution in the double scaling limit of matrix models: a Coulomb fluid approach, J. Phys. A, 28, L207-L211 (1995) · Zbl 0827.15043 · doi:10.1088/0305-4470/28/7/001 |
| [17] | Dai, D., Xu, S.-X., Zhang, L.: On the deformed Pearcey determinant. arXiv:2007.12691 |
| [18] | Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes 3, New York University (1999) · Zbl 0997.47033 |
| [19] | Deift, P.; Its, A.; Krasovsky, I., Asymptotics of the Airy-kernel determinant, Commun. Math. Phys., 278, 643-678 (2008) · Zbl 1167.15005 · doi:10.1007/s00220-007-0409-x |
| [20] | Deift, P.; Its, A.; Krasovsky, I.; Zhou, X., The Widom-Dyson constant for the gap probability in random matrix theory, J. Comput. Appl. Math., 202, 26-47 (2007) · Zbl 1116.15019 · doi:10.1016/j.cam.2005.12.040 |
| [21] | Deift, P.; Its, A.; Zhou, X., 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. Math., 146, 149-235 (1997) · Zbl 0936.47028 · doi:10.2307/2951834 |
| [22] | Deift, P.; Krasovsky, I.; Vasilevska, J., Asymptotics for a determinant with a confluent hypergeometric kernel, Int. Math. Res. Not. IMRN, 2011, 2117-2160 (2011) · Zbl 1216.33013 |
| [23] | Deift, P.; Zhou, X., A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. Math., 137, 295-368 (1993) · Zbl 0771.35042 · doi:10.2307/2946540 |
| [24] | Deschout, K.: Multiple orthogonal polynomial ensembles. Ph.D. Thesis, KU Leuven (2012) |
| [25] | Ehrhardt, T., The asymptotics of a Bessel-kernel determinant which arises in random matrix theory, Adv. Math., 225, 3088-3133 (2010) · Zbl 1241.47059 · doi:10.1016/j.aim.2010.05.020 |
| [26] | Ehrhardt, T., Dyson’s constant in the asymptotics of the Fredholm determinant of the sine kernel, Commun. Math. Phys., 262, 317-341 (2006) · Zbl 1113.82030 · doi:10.1007/s00220-005-1493-4 |
| [27] | Erdős, L.; Krüger, T.; Schröder, D., Cusp universality for random matrices I: local Law and the complex Hermitian case, Commun. Math. Phys., 378, 1203-1278 (2020) · Zbl 1475.60013 · doi:10.1007/s00220-019-03657-4 |
| [28] | Forrester, PJ, Log-Gases and Random Matrices, London Mathematical Society Monographs Series, 34 (2010), Princeton: Princeton University Press, Princeton · Zbl 1217.82003 |
| [29] | Forrester, PJ, The spectrum edge of random matrix ensembles, Nucl. Phys. B, 402, 709-728 (1993) · Zbl 1043.82538 · doi:10.1016/0550-3213(93)90126-A |
| [30] | Geudens, D.; Zhang, L., Transitions between critical kernels: from the tacnode kernel and critical kernel in the two-matrix model to the Pearcey kernel, Int. Math. Res. Not. IMRN, 2015, 5733-5782 (2015) · Zbl 1341.60127 · doi:10.1093/imrn/rnu105 |
| [31] | Hachem, W.; Hardy, A.; Najim, J., Large complex correlated Wishart matrices: fluctuations and asymptotic independence at the edges, Ann. Probab., 44, 2264-2348 (2016) · Zbl 1346.15035 · doi:10.1214/15-AOP1022 |
| [32] | Hachem, W.; Hardy, A.; Najim, J., Large complex correlated Wishart matrices: the Pearcey kernel and expansion at the hard edge, Electron. J. Probab., 21, 36 (2016) · Zbl 1336.15016 · doi:10.1214/15-EJP4441 |
| [33] | Its, AR; Izergin, AG; Korepin, VE; Slavnov, NA, Differential equations for quantum correlation functions, Int. J. Mod. Phys. B, 4, 1003-1037 (1990) · Zbl 0719.35091 · doi:10.1142/S0217979290000504 |
| [34] | Krasovsky, I.: Large Gap Asymptotics for Random Matrices. In: New Trends in Mathematical Physics, XVth International Congress on Mathematical Physics. Springer, pp. 413-419 (2009) · Zbl 1184.15025 |
| [35] | Krasovsky, I., 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, 2004, 1249-1272 (2004) · Zbl 1077.60079 · doi:10.1155/S1073792804140221 |
| [36] | Kuijlaars, ABJ; McLaughlin, KT-R; Van Assche, W.; Vanlessen, M., The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on \([-1,1]\), Adv. Math., 188, 337-398 (2004) · Zbl 1082.42017 · doi:10.1016/j.aim.2003.08.015 |
| [37] | Mehta, ML, Random Matrices (2004), Amsterdam: Elsevier/Academic Press, Amsterdam · Zbl 1107.15019 |
| [38] | Miyamoto, T., On an Airy function of two variables, Nonlinear Anal., 54, 755-772 (2003) · Zbl 1040.33001 · doi:10.1016/S0362-546X(03)00102-0 |
| [39] | Okounkov, A.; Reshetikhin, N., Random skew plane partitions and the Pearcey process, Commun. Math. Phys., 269, 571-609 (2007) · Zbl 1115.60011 · doi:10.1007/s00220-006-0128-8 |
| [40] | Olver, F.W.J., Olde Daalhuis, A.B., Lozier, D.W., Schneider, B.I., Boisvert, R.F., Clark, C.W., Miller, B.R., Saunders, B.V. (eds).: NIST Digital Library of Mathematical Functions, Release 1.0.21 of 2018-12-15. http://dlmf.nist.gov/ |
| [41] | Pastur, LA, The spectrum of random matrices, Teoret. Mat. Fiz., 10, 102-112 (1972) |
| [42] | Pearcey, T., The structure of an electromagnetic field in the neighborhood of a cusp of a caustic, Philos. Mag., 37, 311-317 (1946) · doi:10.1080/14786444608561335 |
| [43] | Tracy, C.; Widom, H., The Pearcey process, Commun. Math. Phys, 263, 381-400 (2006) · Zbl 1129.82031 · doi:10.1007/s00220-005-1506-3 |
| [44] | Zinn-Justin, P., Random Hermitian matrices in an external field, Nucl. Phys. B, 497, 725-732 (1997) · Zbl 0933.82022 · doi:10.1016/S0550-3213(97)00307-6 |
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