×

Universal results for correlations of characteristic polynomials: Riemann-Hilbert approach. (English) Zbl 1098.82018

Summary: We prove that general correlation functions of both ratios and products of characteristic polynomials of Hermitian random matrices are governed by integrable kernels of three different types: a) those constructed from orthogonal polynomials, b) those constructed from Cauchy transforms of the same orthogonal polynomials, and finally c) those constructed from both orthogonal polynomials and their Cauchy transforms. These kernels are related with the Riemann-Hilbert problem for orthogonal polynomials. For the correlation functions we obtain exact expressions in the form of determinants of these kernels. Derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via the Deift-Zhou steepest-descent/stationary phase method for Riemann-Hilbert problems, and in particular to find negative moments of characteristic polynomials. This reveals the universal parts of the correlation functions and moments of characteristic polynomials for an arbitrary invariant ensemble of \(\beta=2\) symmetry class.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
11M41 Other Dirichlet series and zeta functions
11Z05 Miscellaneous applications of number theory
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Andreev, A.V., Simons, B.D.: Correlators of spectral determinants in quantum chaos. Phys. Rev. Lett. 75, 2304–2307 (1995) · doi:10.1103/PhysRevLett.75.2304
[2] Balantekin, A.B., Cassak, P.: Character expansions for the orthogonal and symplectic groups. J. Math. Phys. 43(1), 604–620 (2002) · Zbl 1052.22017 · doi:10.1063/1.1418014
[3] Basor, E.L., Forrester, P.J.: Formulas for the evaluation of Toeplitz determinants with rational generating functions. Math. Nach. 170, 5–18 (1994) · Zbl 0813.15003 · doi:10.1002/mana.19941700102
[4] Berry, M.V., Keating, J.P.: Clusters of near-degenerate levels dominate negative moments of spectral determinants. J. Phys. A: Math. Gen. 35, L1–L6 (2002) · Zbl 1012.81015
[5] Bleher, P., Its, A.R.: Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problems, and universality in the matrix model. Ann. Math. 150, 185–266 (1999) · Zbl 0956.42014 · doi:10.2307/121101
[6] Brézin, E., Hikami, S.: Characteristic polynomials of random matrices. Commun. Math. Phys. 214, 111–135 (2000) · Zbl 1042.82017 · doi:10.1007/s002200000256
[7] Brézin, E., Hikami, S.: Characteristic polynomials of random matrices at edge singularities. Phys. Rev. E 62(3), 3558–3567 (2000) · Zbl 1042.82017 · doi:10.1103/PhysRevE.62.3558
[8] Brézin, E., Hikami, S.: New correlation functions for random matrices and integrals over super groups. arXiv:math-ph/0208001, 2002 · Zbl 1066.82022
[9] Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinshtein, M.O., Snaith, N.C.: Autocorrelation of Random Matrix Polynomials. arXiv: mat.nt/0208007, 2002
[10] Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinshtein, M.O., Snaith, N.C.: Integral moments of zeta- and L-functions. arXiv: mat.nt/0206018, 2002
[11] Deift, P.: Integrable operators. In: Differential operators and spectral theory: M. Sh. Birman’s 70th anniversary collection, V. Buslaev, M. Solomyak, D. Yafaev, eds., American Mathematical Society Translation, Ser.2, V. 189, Providence, RI: Am. Math. Soc., 1999, pp. 69
[12] Deift, P.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes, 3 New York: Courant Institute of Mathematical Sciences, New York University, 2000 · Zbl 0997.47033
[13] Deift, P., Its, A.R., Zhou, X.A.: 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. (2) 146, 149–235 (1997) · Zbl 0936.47028
[14] Deift, P., Zhou, X.A.: A steepest descent method for oscillatory Riemann-Hilbert problem, Asymp- totics for the MKdV equation. Ann. Math. (2) 137, 295–368 (1993) · Zbl 0771.35042
[15] Deift, P., Zhou, X.A.: Asymptotics for the Painlevé II equation. Commun. Pure Appl. Math. 48, 277–337 (1995) · Zbl 0869.34047 · doi:10.1002/cpa.3160480304
[16] Deift, P., Venakides, S., Zhou, X.A.: The collisionless shock region for the long-time behavior of solutions of the KdV equation. Commun. Pure Appl. Math. 47(2), 199–206 (1994) · Zbl 0797.35143 · doi:10.1002/cpa.3160470204
[17] Deift, P., Kriecherbauer, T., McLaughlin, K.T-R., Venakides, S., Zhou, X.A.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52(12), 1491–1552 (1999) · Zbl 1026.42024 · doi:10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.0.CO;2-#
[18] Deift, P., Kriecherbauer, T., McLaughlin, K.T-R., Venakides, S., Zhou, X.A.: Uniform asymp- totics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52(11), 1335–1425 (1999) · Zbl 0944.42013 · doi:10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1
[19] Efetov, K.B.: Supersymmtry in Disorder and Chaos. Cambridge: Cambridge University Press, 1997 · Zbl 0990.82501
[20] Fokas, A.S., Zaharov, V.E., eds.: Important developments in soliton theory. Berlin: Springer-Verlag, 1993 · Zbl 0801.00009
[21] Fokas, A.S., Its, A.R., Kitaev, A.V.: An isomondromy approach to the theory of two-dimensional quantum gravity. Russ. Math. Surv. 45(6), 155–157 (1990) · Zbl 0743.35055
[22] Fokas, A.S., Its, A.R., Kitaev, A.V.: Discrete Painlevé equations and their appearance in quantum gravity. Commun. Math. Phys 142(2), 313–344 (1991) · Zbl 0742.35047 · doi:10.1007/BF02102066
[23] Forrester, P.J., Witte, N.S.: Application of the tau-function theory of Painlevé equations to random matrices: PIV, PII and the GUE. Commun. Math. Phys. 219(2), 357–398 (2001) · Zbl 1042.82019 · doi:10.1007/s002200100422
[24] Forrester, P.J., Witte, N.S.: Commun. Pure Appl. Math. 55, 679–727 (2002) · Zbl 1029.34087 · doi:10.1002/cpa.3021
[25] Fyodorov, Y.V.: Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation. Nucl. Phys. B 621, 643–674 (2002) · Zbl 1024.82013 · doi:10.1016/S0550-3213(01)00508-9
[26] Fyodorov, Y.V., Keating, J.P.: Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations. J. Phys. A.: Math. Gen. 36, 4035–4046 (2003) · Zbl 1049.82035 · doi:10.1088/0305-4470/36/14/308
[27] Fyodorov, Y.V., Khoruzhenko, B.A.: Systematic Analytic Approach to Correlation Fuctions of Resonances in Quantum Chaotic Scattering. Phys. Rev. Lett. 83, 65–68 (1999) · doi:10.1103/PhysRevLett.83.65
[28] Fyodorov, Y.V., Sommers, H.-J.: Universality of ”level curvature” distribution for large random matrices: Systematic analytical approaches. Eur. Phys. J. B 99(1), 123–135 (1995)
[29] Fyodorov, Y.V., Sommers, H-J: Random Matrices close to Hermitian or unitary: Overview of methods and results. J. Phys. A: Math. Gen. 36, 3303–3347 (2003) · Zbl 1069.82006 · doi:10.1088/0305-4470/36/12/326
[30] Fyodorov, Y.V., Strahov, E.: Characteristic polynomials of random Hermitian matrices and Duistermaat-Heckman localisation on non-compact Kähler manifolds. Nucl. Phys. B 630, 453–491 (2002) · Zbl 0994.15025 · doi:10.1016/S0550-3213(02)00185-2
[31] Fyodorov, Y.V., Strahov, E.: On correlation functions of characteristic polynomials for chiral Gaussian Unitary Ensemble. Nucl. Phys. B 647, 581–597 (2002) · Zbl 1001.15016 · doi:10.1016/S0550-3213(02)00904-5
[32] Fyodorov, Y.V., Strahov, E.: An exact formula for general spectral correlation function of random Hermitian matrices. J. Phys. A: Math. Gen. 36, 3203–3213 (2003) · Zbl 1044.81050 · doi:10.1088/0305-4470/36/12/320
[33] Gonek, S.M.: On negative moments of the Riemann zeta-fuction. Mathematika 36, 71–88 (1989) · Zbl 0673.10032 · doi:10.1112/S0025579300013589
[34] Haake, F.: Quantum Signatures of Chaos. 2nd ed., Berlin-Heidelberg-New York: Springer, 2000 · Zbl 1209.81002
[35] Hua, L.K.: Harmonic Analysis of Functions of Several Complex Variables in Classical Domains. Providence, RI: 1963 American Mathematical Society · Zbl 0112.07402
[36] Hughes, C.P., Keating, J.P., O’Connell, N.: Random matrix theory and the derivative of the Riemann zeta function. Proc. R. Soc. Lond. A 456, 2611–2627 (2000) · Zbl 0996.11052 · doi:10.1098/rspa.2000.0628
[37] Hughes, C.P., Keating, J.P., O’Connell, N.: On the characteristic polynomials of a random unitary matrix. Commun. Math. Phys. 220(2), 429–451 (2001) · Zbl 0987.60039 · doi:10.1007/s002200100453
[38] Its, A.R., Izergin, A.G., Korepin, V.E.: Temperature correlators of the impenetrable Bose gas as an integrable system. Commun. Math. Phys 129, 205–222 (1990) · Zbl 0698.60094 · doi:10.1007/BF02096786
[39] Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: Differential equations for quantum correlation functions. Int. J. Mod. Phys. B4, 1003–1037 (1990) · Zbl 0719.35091
[40] Its, A.R., Izergin, A.G., Korepin, V.E., Slavnov, N.A.: The quantum correlation function as the {\(\tau\)} function of classical differential equations. In: Important developments in soliton theory, A.S. Fokas and V.E. Zakharov, eds., Berlin: Springer-Verlag, 1993, pp. 407–417 · Zbl 0814.35108
[41] Johansson, K.: On fluctuations of eigenvalues of random hermitian matrices. Duke Math. J. 91(1), 151–204 (1998) · Zbl 1039.82504 · doi:10.1215/S0012-7094-98-09108-6
[42] Kanzieper, E.: Replica field theories, Painlevé transcendents and exact correlation functions. Phys. Rev. Lett. 89(25), 250201 (2002) · Zbl 1267.81236 · doi:10.1103/PhysRevLett.89.250201
[43] Keating, J.P., Snaith, N.C.: Random matrix theory and {\(\zeta\)}(1/2+it). Commun. Math. Phys. 214, 57–89 (2000) · Zbl 1051.11048 · doi:10.1007/s002200000261
[44] Mehta, M.L.: Random Matrices. New York: Academic Press, 1991 · Zbl 0780.60014
[45] Mehta, M.L., Normand, J-M.: Moments of the characteristic polynomial in the three ensembles of random matrices. J. Phys. A: Math. Gen. 34, 4627–4639 (2001) · Zbl 1129.82309 · doi:10.1088/0305-4470/34/22/304
[46] Mirlin, A.D., Fyodorov, Y.V.: Universality of level correlation function of sparse random matrices. J. Phys. A: Math. Gen. 24(10), 2273–2286 (1991) · Zbl 0760.15017 · doi:10.1088/0305-4470/24/10/016
[47] Nonnenmacher, S., Zirnbauer, M.R.: Det-Det Correlations for quantum maps: Dual pair and saddle-point analyses. J. Math. Phys. 43(5), 2214–2240 (2002) · Zbl 1059.81068 · doi:10.1063/1.1462417
[48] Sagan, B.: The Symmetry Group Representations, Combinatorial Algorithms, and Symmetric Functions. New York: Springer, 2000
[49] Splittorff, K., Verbaarschot, J.J.M.: Replica Limit of the Toda Lattice Equation. Phys. Rev. Lett. 90(4), 041601 (2003) · Zbl 1267.82071 · doi:10.1103/PhysRevLett.90.041601
[50] Szabo, R.J.: Microscopic spectrum of the QCD Dirac operator in three dimensions. Nucl. Phys. B 598(1–2), 309–347 (2001) · Zbl 1046.81554
[51] Szegö, G.: Orthogonal polynomials. Colloquium Publications, 23, Providence, RI: AM. Math. Soc., 1975 · Zbl 0305.42011
[52] Tracy, C.A., Widom, H.: Introduction to random matrices. Geometric and quantum aspects of integrable systems. Lecture Notes in Phys., Vol. 424, Berlin: Springer, 1993, pp. 103–130 · Zbl 0791.15017
[53] Tracy, C.A., Widom, H.: Level spacing distribution and the Airy kernel. Commun. Math. Phys. 159, 151–174 (1994) · Zbl 0789.35152 · doi:10.1007/BF02100489
[54] Tracy, C.A., Widom, H.: Level spacing distribution and the Bessel kernel. Commun. Math. Phys. 161, 289–309 (1994) · Zbl 0808.35145 · doi:10.1007/BF02099779
[55] Tracy, C.A., Widom, H.: Fredholm determinants, differential equations and matrix models. Commun. Math. Phys. 163, 33–72 (1994) · Zbl 0813.35110 · doi:10.1007/BF02101734
[56] Zirnbauer, M.R.: Dual pairs in random matrix theory. Talk given at LMS Workshop: Zeta Functions, Random Matrices and Quantum Chaos. September 13–14, 2001
[57] Zirnbauer, M.R.: Random Matrices, Symmetry Classes, and Dual Pairs. Talk given at James H. Simons Workshop on Random Matrix Theory. Stony Brook, February 20–23, 2002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.